Some Recent Results on Compact Operators and Weakly Compact Operators

We start with the following: 19.1 Definition.

Let E and F be normed spaces and T ∈ L(E,F). We say that

T is:

a finite operator (or an operator of finite rank) if dim T ( E ) < ∞ ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3153.tif"/>

compact (resp. precompact) if T(U_E) is relatively compact(resp. totally bounded) in F;

weakly compact if T(U_E) is relatively σ(F, F^')–compact in F;

completely continuous if every σ(E, E^')–convergent sequence {x_n} in E is mapped into a norm convergent sequence {Tx_n} in F.

Denote by F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3154.tif"/> (E,F) (resp. L https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3155.tif"/> ^p(E,F), L https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3156.tif"/> ^c(E,F), L https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3157.tif"/> ^wc(E,F), and L https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3158.tif"/> ^cc(E,F)) the vector space of all finite operators (resp. precompact, compact, weakly compact and completely continuous operators) from E into F.

Remarks

(i) It is clear that F ( E,F ) ⊂ L c ( E,F ) ⊂ L p ( E,F ) , L c ( E,F ) ⊂ L wc ( E,F )      and    L c ( E,F ) ⊂ L cc ( E,F ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3159.tif"/>

Moreover, if F is complete then L https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3160.tif"/> ^c(E,F) = L https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3161.tif"/> ^p(E,F).

(ii) As the unit ball U_E in E is a 0–neighbourhood as well as a bounded set, the notion of compact (resp. weakly compact) can be generalized to the case of LCS in two directions from the global (i.e., neighbourhoods) and local (i.e., homologies) property as follows: Let (X, P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3162.tif"/> ) and (Y, J https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3163.tif"/> ) be LCS and T : X → Y a linear map. We say that 247T is:

precompact (resp. compact, weakly compact and bounded) if there exists a P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3164.tif"/> –neighbourhood V of 0 in X such that T(V) is totally bounded (resp. relatively P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3165.tif"/> –compact, relatively σ(Y,Y^')–compact and bounded) in Y.

locally precompact (resp. locally compact, locally weakly compact and locally bounded) if for any P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3166.tif"/> –bounded subset B of X, T(B) is precompact (resp. relatively J https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3167.tif"/> –compact, relatively σ(Y,Y^')–compact and bounded) in Y.

Denote by L L p ( X , Y ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3168.tif"/> (resp. L L c ( X , Y ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3169.tif"/> , L L wc ( X , Y ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3170.tif"/> , and L L b ( X , Y ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3171.tif"/> ) the vector space of all precompact (resp. compact, weakly compact and bounded) operators from X into Y, and by L^lp(X,Y) (resp. L^lc(X,Y), L^lwc(X,Y), and L^lb(X,Y) (or L https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3172.tif"/> ^x(X,Y))) the vector space of all locally precompact (resp. locally weakly compact and locally bounded) operators from X into Y. Then we have : L L c ( X,Y ) ⊂ L L p ( X,Y ) ∩ L L wc ( X,Y ) ⊂ L L b ( X,Y ) ⊂ L ( X,Y ) ; L L b ( X,Y ) = L ( E,F )    whenever    Y    is    normble;    and L L α ( X,Y ) ⊂ L 1 α ( X,Y )    whenever     α    represents    c, p,    wc    and    b . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3173.tif"/>

Moreover, the composite of two operators in which one of them belongs to is in L L α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3174.tif"/> is in L L α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3175.tif"/> (where α represents c, p, wc and b).

19.a Finite representation:

Let E and F be normed spaces and T ∈ L(E,F). Then T is a finite operator if and only if T is of the form (19.a.1) Tx = ∑ i = 1 n < x ,   u ′ i > y i   (for all x ∈ E )  with  u ′ i ∈ E '  and y i ∈ F  , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3176.tif"/>

thus we write T = ∑ i = 1 n u ′ i ⊗ y i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3177.tif"/> . Of course, the above finite representation (19.a.1) is not unique, but it can be assumed that both sets { u ′ i : i = 1 , ⋯ , n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3178.tif"/> and { y i : i = 1 , ⋯ , n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3179.tif"/> are linearly independent sets of n elements. The number n is uniquely determined by T and 248called the rank of T (see Schaefer [1966, EX. (Ill IS) p. 119]).

19.b Compact operators between Banach spaces:

Let E and F be Banach spaces. Then L https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3180.tif"/> ^c(E,F) is a closed vector subspace of ( L ( E , F ) , | | ⋅ | | ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3181.tif"/> . Moreover, the composite of operators in which one of them is compact must be compact.

Finite operators are compact operators, the converse is, in general, not true. However we have the following interesting result.

19.2 Proposition

Let E and F be Banach spaces and T ∈ L(E,F) a compact operator. If T(E) is closed in F then T is a finite operator.

Proof.

The closedness of T(E) in F, together with the completeness of F, shows that T is an open map from E onto the Banach space T(E) (by Banach’s open mapping theorem), hence T(U_E) is a relative 0–neighbourhood in T(E) which is relatively compact (by the assumption on the compactness of T), thus dim T(E) < ∞.

The first duality theorem for compact operators is the following:

19.3 Theorem (Schauder)

Let E and F be Banach spaces and T ∈ L(E,F). Then T is compact if and only if T^' : F^' → E^' is compact.

Proof.

Necessity. We shall employ Arzela–Ascoli’s theorem to prove that T^'(U_F′) is relatively sequentially compact in the Banach space E^' or, equivalently, for any sequence { y ′ n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3182.tif"/> in U_F′ there exists a subsequence { y ′ n k } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3183.tif"/> of { y ′ n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3184.tif"/> and u′ ∈ E^' such that (1) lim k − ∞   ‖ T ' y ′ n k − u ′ ‖ = 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3185.tif"/>

In fact, by assumption, K = T ( U E ) ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3186.tif"/> is compact in F. On the other hand, U F ′ ⊂ ( C ( K ) , | | ⋅ | | ∞ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3187.tif"/> has the following properties:

249(i) U_F' is uniformly bounded on K (i.e. U_F', is bounded in the Banach space ( C ( K ) , | | ⋅ | | ∞ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3188.tif"/> . Indeed, for any x ∈ U_E with z = Tx ∈ T(U_E), | g ( z ) | ≤ ‖ g ‖   ‖ z ‖ = ‖ g ‖   ‖ Tx ‖ ≤ ‖ g ‖   ‖ T ‖   ‖ x ‖                           ≤ ‖ T ‖    ( for    all    g ∈ U F ' ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3189.tif"/>

hence sup g ∈ U F '   ‖ g ‖ ∞ = sup   { sup z ∈ K   | g ( z ) | : g ∈   U F ' } ≤ ‖ T ‖   . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3190.tif"/>

(ii) U_F' an equicontinuous subset of C(K): For any g ∈ U_F' we have | g ( z 1 ) − g ( z 2 ) | ≤ ‖ g ‖   ‖ z 1 − z 2 ‖                                                               ≤ ‖ z 1 − z 2 ‖    ( for    all    z 1 ,   z 2   ∈   T ( U E ) ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3191.tif"/>

thus U_F' is equicontinuous.

By Arzelá–Ascoli’s theorem, U_F' is relatively compact in ( C ( K ) , | | ⋅ | | ∞ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3192.tif"/> , thus for any sequence { y ′ n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3193.tif"/> in U_F', there exists a subsequence { y ′ n k } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3194.tif"/> of { y ′ n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3195.tif"/> such that sup x ∈ U E   | y ′ n k ( Tx ) − y ′ n j ( Tx ) | →   0   ( as     k, j →   ∞ ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3196.tif"/>

It then follows that     ‖ T ' y ′ n k − T y ′ n j ‖ = sup x ∈ U E   { | T ' y ′ n k ( x ) − T ' y ′ n j ( x ) | } = sup x ∈ U E { | y ′ n k ( Tx ) − y ′ n j ( Tx ) | } →   0   ( as    k,j →   ∞ ) ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3197.tif"/>

in other words, { T ' y n k ' } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3198.tif"/> is a Cauchy sequence in the Banach dual E^', thus there is an u^' ∈ E^' such that lim k   | | T ' y ' n k − u ' | | = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3199.tif"/> . Therefore T^' is compact.

Sufficiency

We first observe that K E U E ⊂ U E ' ' https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3200.tif"/> , it then follows from 250 K F T ⊂ T ' ' K E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3201.tif"/> that K F T ( U E ) = T ' ' K E ( U E ) ⊂ T ' ' ( U E ' ' ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3202.tif"/>

Suppose now that T^' is compact. Then T ' ' : E ' ' → F ' ' https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3203.tif"/> is compact (by the first half of the theorem), thus T ' ' ( U E ' ' ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3204.tif"/> is totally bounded in F”; consequently, K_F(TU_E) is totally bounded in F ' ' https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3205.tif"/> . As K F : F → F ' ' https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3206.tif"/> is a metric injection, we conclude that TU_E is totally bounded in the Banach space F, and hence that T is compact.

Remark

Let E and F be normed spaces and T ∈ L(E,F). One can show that T is precompact if and only if T^' ∈ L https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3207.tif"/> ^c(F′,E′).

19.c A generalization of Schauder’s theorem(see <xref ref-type="bibr" rid="ref65">Köthe [1979, p.202]</xref>):

Let X and Y be F–spaces and T ∈ L(X,Y). Then T is locally compact if and only if T ' ∈ L ( Y β ' , X β ' ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3208.tif"/> is locally compact.

Using Schauder’s duality theorem for compact operators and Grothendieck’s structure theorem for compact sets in Banach spaces, we are able to verify the following remarkable result.

19.4 Theorem (<xref ref-type="bibr" rid="ref109">Terzioglu [1971]</xref>,<xref ref-type="bibr" rid="ref88">Randtke [1972]</xref>)

Let E and F be Banach spaces and T ∈ L(E,F). Then the following statements are equivalent:

T is compact.

There exists a null sequence {f_n} in E^' such that (1) ‖ Tx ‖ ≤ sup   n | < x,f n > |        ( for    all    _ x ∈   E ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3209.tif"/>

There exists an [λ_n] ∈ c₀ and an equicontinuous sequence { u ′ n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3210.tif"/> in E^' such that ‖ Tx ‖ ≤ sup n   | λ n < x,  u ′ n > |       ( for    all    x  ∈  E ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3211.tif"/>

251 There exists a closed subspace H of c₀, R ∈ L https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3212.tif"/> ^c(E,R) and S ∈ L(H,F) such that T = SR.

Proof.

(a) ⇒ (b): By (19.3), T^' :F^' → E^' is compact, hence T^'(U_F′) is relatively compact in E^', thus Grothendieck’s structure theorem for compact sets (see (5.2)) shows that there exists a null sequence {f_n} in E^' such that each element u′ ∈ T^'(U_F′) has a representation u ′   = ∑ n = 1 ∞ μ n f n    with     ∑ n | μ n | ≤ 1. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3213.tif"/>

For any x ∈ E, we have ‖ Tx ‖ = sup   { | < Tx , y ′ > | : y ′ ∈ U F ′ } = sup   { | < x, T ' y ′ > | : y ′ ∈ U F ′ }       ≤ ∑ k = 1 ∞ | μ k | sup n   | < x, f n > | ≤ sup n   | < x, f n > | . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3214.tif"/>

(b) ⇒ (c): Trivial.

(b) ⇒ (d): Suppose that {f_n} is a null sequence in E^' such that (1) holds. Then {f_n} is bounded, σ(E^',E)–null sequence, hence the map R : E → c₀, defined by, Rx   = [ < x, f n > ] n ≥ 1 = Σ n < x, f n > e n      ( for    all    x  ∈    E ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3215.tif"/>

is an operator with | | R | | = sup n   | | f n | | https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3216.tif"/> such that (2) ‖ Tx ‖ ≤ sup   n | < x,f n > | = ‖ Rx ‖ ∞    ( for    all    x    ∈     E ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3217.tif"/>

[see(3.j) (ii)]. It is not hard to show that the set, defined by K = { [ ζ n ] ∈ c 0 : | ζ n | ≤ ‖ f n ‖      ( for    all    n  ≥  1 ) } , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3218.tif"/>

252is compact in c₀ (since | | f n | | → 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3219.tif"/> . As R(U_E) ⊂ K, it follows that R ∈ L https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3220.tif"/> ^c(E,c₀).

On the other hand, (2) ensures that Ker R ⊂ Ker T, hence there is an operator S : R(E) → F such that (3) T = SR    and    ‖ SR ( x ) ‖ = ‖ Tx ‖ ≤ ‖ Rx ‖ ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3221.tif"/> (by (1) and(2)). Let H be the closure of R(E) in c₀. Then R: E → H is still compact. On the other hand, the completeness of F ensures that S ∈ L(R(E), F) has a unique continuous extension to H, which is denoted again by S. By (3), we have T = SR.

(d) ⇒ (a): Trivial.

Remark

If E and F are assumed only normed spaces, one can show that T is precompact if and only if the statement (b) (or (c)) is true.

We say that an operator T ∈ L(E,F) admits a compact factorization through a Banach space (resp. a LCS) Z if there are compact operators R ∈ L(E,Z) and S ∈ L(Z,Y) such that T = SR . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3222.tif"/>

As an immediate consequence of (19.4), we obtain the following:

19.5 Corollary (<xref ref-type="bibr" rid="ref42">Figiel [1973]</xref>)

Let E and F be Banach spaces. An operator T ∈ L(E,F) admits a compact factorization through a Banach space if and only if it admits a compact factorization through a closed subspace of c₀.

From the equivalence of (a) and (d) of (19.4), it is natural to ask in what cases a compact operator T : E → F admits a compact factorization through the whole space c₀. The answer is given by Terzioglu [1972], as stated by the following:

19.6 Theorem (<xref ref-type="bibr" rid="ref110">Terzioglu [1972]</xref>)

253 Let E and F be Banach spaces andT ∈ L(E,F) a compact operator. Then T admits a compact factorization through the whole space c₀ if and only if T is an ∞–nuciear operator in the following sense: there exists an [ζ_n] ∈ c₀, an equicontinuous sequence {f_n} in E^' and a summable sequence {y_n} in F [for definition see (3.h)] such that Tx   =   Σ n ζ n < x,    f n >   y n    ( for    all    x   ∈   E ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3223.tif"/>

For a proof, see Terzioglu [1972] or Köthe [1979, p.277].

19.d Precompact operators between LCS (Randtke [<xref ref-type="bibr" rid="ref88">1972</xref>, (<xref ref-type="statement" rid="lemma2_10">2.10</xref>)] and <xref ref-type="bibr" rid="ref124">Wong [1979, (1.1.2) and (1.1.6)]</xref>):

Let X and Y be LCS, and T ∈ L(X,Y). If there exist an [ζ_n] ∈ c₀, an equicontinuous sequence {f_n} in X^' and a bounded disk B in Y such that (d.1) r B ( Tx ) ≤ sup n | ζ n < x , f n > |    ( for    all    x  ∈  X ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3224.tif"/> (where r_B is the gauge of B defined on Y(B)), then T is precompact.

The converse is true provided that Y is metrizable; in this case (i.e., Y is metrizable), it is equivalent to the following:

(*) There exist normed spaces E and F, a precompact operator T ˜ ∈ L ( E , F ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3225.tif"/> and operators Q ∈ L(X,E) and J ∈ L(E,Y) such that T = J T ˜ Q https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3226.tif"/> .

In order to extend the results of (19.4) and (19.6), we require the following terminology and results.

19.7 Definition.

Let X and Y be LCS. A seminorm p on X is said to be precompact if there exists an [ζ_n] ∈ c₀ and an equicontinuous sequence {f_n} in X^' such that p ( x ) ≤ sup n   | ζ n < x,f n > |      ( for    all    x  ∈    X ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3227.tif"/>

254We say that an operator T ∈ L(E,F) is:

a quasi–Schwartz (or precompact–bounded in the terminology of Wong [1979, p.24]) if there exists a precompact seminorm p on X such that { Tx : p ( x ) ≤ 1 } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3228.tif"/> is bounded in Y;

b–precompact if there exists a 0–neighbourhood V in X such that T(V) is b–precompact in Y (for definition, see (15.2)(a)).

We denote by L L pb ( X , Y ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3229.tif"/> the vector space of all b–precompact operators form X into Y.

Remarks

(i) The notions of precompact seminorms and quasi-Schwartz operators are due to Randtke [1972], while the concept of b–precompact operators is due to Wong [1982a].

(ii) It is clear that every precompact seminorm on a LCS must be continuous.

(iii) The composite of two operators in which one of them is quasi–Schwartz (resp. b–precompact) is quasi–Schwartz (resp. b–precompact). Moreover, L L pb ( X , Y ) ⊂ L L p ( X , Y ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3230.tif"/> , and they are equal when Y is metrizable [see (11.2) and (15.2) (a)]. Thus the notion of b–precompact operators is a natural generalization of that of precompact operators. We shall show (below (19.8)) that b–precompact operators are exactly quasi-Schwartz operators.

19.e Characterization of precompact seminorms (Randtke [<xref ref-type="bibr" rid="ref88">1972</xref>, (<xref ref-type="statement" rid="definition2_4">2.4</xref>)]):

Let (X, P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3231.tif"/> ) be a LCS, let p be a continuous seminorm on X, and let X_p be the quotient space X/p^-1(0) equipped with the quotient norm p of p by p^-1(0). Then the following statements are equivalent.

p is precompact.

Q_p : (X, P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3232.tif"/> ) → X_p is a precompact operator.

255There exists a continuous seminorm q on X with p ≤ q such that the canonical map Q_p,q : X_q → X_p is precompact.

19.8 Theorem (<xref ref-type="bibr" rid="ref88">Randtke [1972]</xref>, <xref ref-type="bibr" rid="ref127">Wong [1982a]</xref>)

Let X and Y be LCS and T ∈ L(X,Y). Then the following statements are equivalent.

T is quasi–Schwartz.

There exist normed spaces E and F, a precompact operator T ˜ ∈ L ( E , F ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3233.tif"/> and operators Q ∈ L(X,E) and J ∈ L(F, Y) such that T = J T ¯ Q . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3234.tif"/>

T is a b–precompact operator.

T admits a quasi–Schwartz factorization through a vector subspace H of c₀ in the following sense: there exists quasi–Schwartz operators T₁ ∈ L(X, H) and T₂ ∈ L(H, Y) such that T = T 2 T 1 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3235.tif"/>

Proof.

(a) ⇒ (b): Let p be a precompact seminorm on X such that B = { Tx : p ( x ) ≤ 1 } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3236.tif"/> is a bounded disk in Y and V = { x ∈ X : p ( x ) < 1 } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3237.tif"/> . Then p^-1(0) ⊂ Ker T [by the boundedness of B], hence there is an S∈L(X_p,Y(B))(in fact, | | S | | ≤ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3238.tif"/> by T(V) ⊂ B) such that J_BSQ_p = T. Now the precompactness of p ensures [see (19.e)] that there is a continuous seminorm q on X with p ≤ q such that Q_p,q : X_q → X_p is precompact. Therefore we have T = J B SQ p,q Q q    with    T ˜ = SQ p,q ∈ L p ( X q , Y ( B ) ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3239.tif"/>

hence the implication follows.

(b) ⇒ (c): By (11.2), every precompact set in a metrizable LCS (Z, T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3240.tif"/> ) must be b–precompact in (Z, T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3241.tif"/> ), the implication follows from Remark (iii) of (19.7).

(c) ⇒ (d): There exists a disked 0–neighbourhood V in X and a bounded disk B 256in Y such that T(V) is precompact in the normed space Y(B) = (Y(B),γ_B). We claim that the precompact operator T : X → (Y(B),γ_B) admits a precompact factorization through a vector subspace of c₀.

In fact, by (d.1) of (19.d), there exists an [ζ_n] ∈ c₀ and an equicontinuous sequence {f_n} in X^' such that (1) γ B ( Tx ) ≤ sup n   { | ζ n 2 < x,f n > | }    ( for    all    x  ∈  X ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3242.tif"/>

Now we define a map T₁ : X → c₀ by setting (2) T 1 x   = [ ζ n < x,f n > ] n ≥ 1   ( for    all    x  ∈  X ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3243.tif"/>

and let H = T₁X. Then T₁ is precompact operator from X into a vector subspace H of c₀ [see (3.j)(ii)]. The diagonal translation D : c₀ → c₀, defined by (3) D [ ( η n ) ] = [ ζ n η n ] n ≥ 1   ( for    all    [ η n ] ∈ c 0 ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3244.tif"/>

is a compact operator such that Ker ( D°T 1 ) ⊂ Ker T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3245.tif"/> (by (1)), hence there exists a linear map S : D(H) → Y(B) such that T = SDT₁. For any x ∈ X, we have from (2) and (3) that ( DT 1 ) x = [ ζ n 2 < x , f n > ] n ≥ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3246.tif"/> , it then follows from (1) that γ B ( ( SDT 1 ) x ) = γ B ( Tx ) ≤ sup n   { | ζ n 2 < x , f n > | } = ‖ DT 1 x ‖ ∞ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3247.tif"/>

and hence that S is continuous. Consequently, SD : H → Y(B) is a precompact operator. This proves our assertion.

Finally, since H and Y(B) are normed spaces, it follows that T₁ : X → H and SD : H → Y(B) are quasi–Schwartz operators, and hence that T₂ = J_BSD : H → Y is a quasi–Schwartz operator such that T = SDT 1 = ( J B SD ) T 1 = T 2 T 1 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3248.tif"/>

thus T is the composite of two quasi–Schwartz operators T₁ and T₂.

257(d) ⇒ (a): Trivial.

The equivalence of (a) and (b) is due to Randtke [1972.(2.1)], while the equivalences between (a), (c) and (d) are due to Wong [1982a, Theorems 2 and 5].

It is clear that b–precompact sets in a LCS Y are precompact, but the converse is, in general, not true (except for the metrizable case of Y [see (11.2) and (15.2) (a)]). Thus the notion of quasi–Schwartz operators is a natural generalization of that of precompact operators; consequently, the preceding result is a generalization of (19.4).

From (19.8), it is natural to ask in what case a quasi-Schwartz operator has a quasi–Schwartz factorization through the whole space c₀. To answer this question, we require the following:

19.9 Definition (<xref ref-type="bibr" rid="ref88">Randtke[1972]</xref>)

A sequence {y_n} in a LCS Y is said to be strongly summable if it satisfies the following two conditions:

for any [ λ n ] ∈ ℓ ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3249.tif"/> the series ∑ n λ n y n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3250.tif"/> converges in Y;

{ ∑ n λ n y n : | | λ n | | ∞ ≤ 1 , [ λ n ] ∈ ℓ ∞ } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3251.tif"/> is bounded in Y.

Let X be a LCS. An operator T ∈ L(X,Y) is called a Schwartz operator if there exists an [ζ_n] ∈ c₀, an equicontinuous sequence {f_n} in X^' and a strongly summable sequence {y_n} in Y such that Tx = Σ n ζ n < x,f n > y n    ( for    all    x  ∈  X ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3252.tif"/>

Remark

It can be shown (see Wong [1980b, Lemma 1]) that if Y is a sequentially complete LCS and {z_n} is a weakly summable sequence in Y, then for any [λ_n] ∈ c₀, the sequence {λ_nz_n} in Y must be strongly summable. Thus the notion of Schwartz operators is a natural extension of that of ∞–nuclear operators (see (19.6)).

19.10 Theorem (Wong [1980b])

Let X and Y be LCS and T ∈ L(X,Y) a 258 quasi–Schwartz operator. Then T is a Schwartz operator if and only if T admits a quasi–Schwartz factorization through the whole space c₀. (Consequently, Schwartz operators must be compact.)

For a proof of this result, we refer to Wong [1980b, Theorem 3].

Combining (19.8),we see that the preceding result is a generalization of (19.6).

We conclude this section with some criteria for weak compactness of operators.

19.11 Theorem

Let E and F be Banach spaces and T ∈ L(E,F). Then the following statements are equivalent.

T is weakly compact.

T ' : ( F ' , σ ( F ' , F ) ) → ( E ' , σ ( E ' , E'' ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3253.tif"/> is continuous.

T'' ( E'' ) ⊂ K F ( F ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3254.tif"/> .

Proof.

(b) ⇒ (c): Follows (16.4) and (17.11).

(a) ⇒ (c): We first show that if B ⊂ F is relatively σ(F,F^')–compact, then (1) clu σ ( F'',F ' ) K F ( B ) ⊂ K F ( F ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3255.tif"/>

where clu σ ( F'',F ' ) K F ( B ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3256.tif"/> denotes the σ ( F'',F ' ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3257.tif"/> –closure of K_F(B).

In fact, as the σ(F,F^')–closure of B, denoted by B ¯ σ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3258.tif"/> , is σ(F,F^')–compact and K F : ( F , σ ( F , F ' ) ) → ( F'', σ ( F'',F ' ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3259.tif"/> is continuous, it follows that K F ( B ¯ σ ) ( ⊂ K F ( F ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3260.tif"/> is σ ( F'',F ' ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3261.tif"/> –compact in F'' https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3262.tif"/> , and surely σ ( F'',F ' ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3263.tif"/> –closed; consequently, clu σ ( F'',F ' ) K F ( B ) ⊂ clu σ ( F'',F ' ) K F ( B ¯ σ )   = K F ( B ¯ σ )   ⊂ K F ( F ) ,   https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3264.tif"/>

which obtains our assertion (1).

259Now the weak compactness of T implies that T(U_E) is relatively σ(F,F^')–compact, hence (1) and T''K E = K F T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3265.tif"/> imply that (2) clu σ ( F'',F ' ) ( T''K E ) U E = clu σ ( F'',F ' ) K F ( T ( U ε ) ) ⊂ K F ( F ) .   https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3266.tif"/>

Notice that U E'' = ( U E ° ) ° ( E'' ) = clu σ ( E'',E ' ) K E ( U E ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3267.tif"/> and that T'': ( E'', σ ( E'',E ' ) ) →   ( F'' , σ ( F'',F ' ) )    is    continuous, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3268.tif"/>

[this is equivalent to say that T'' ( clu σ ( E'',E ' ) A ) ⊂ clu σ ( F'',F ' ) T''A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3269.tif"/> (for any A ⊂ E'' https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3270.tif"/> )], we then conclude from (2) that T''U E'' ⊂ clu σ ( F'',F ' ) T'' ( K E ( U E ) ) ⊂ K F ( F ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3271.tif"/>

and hence from E'' = ∪ n  nU E'' https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3272.tif"/> that T''E'' ⊂ K F ( F ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3273.tif"/> .

(c) ⇒ (a): By (16.4), T'' : ( E'' , σ ( E'' , E ' ) ) → ( F , σ ( F , F ' ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3274.tif"/> is continuous, hence T'' ( U E'' ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3275.tif"/> is σ(F,F^')–compact (since U E'' https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3276.tif"/> is σ ( E'' , E ' ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3277.tif"/> –compact (by Alaoglu–Bourbaki’s theorem)). On the other hand, since σ ( F , F ' ) = σ ( F'' , F ' ) | F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3278.tif"/> and K F ( TU E ) = T''K E ( U E ) ⊂ T''U E'' ⊂ K F ( F ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3279.tif"/>

we conclude that T(U_E) is relatively σ(F,F^')–compact.

The equivalence of (a) and (c) of (19.11) is due to Gantmacher and Nakamura.

Let E and F be Banach spaces and T ∈ L(E,F). If either E or F is reflexive, then T must be weakly compact. On the other hand, the composite of two operators in which one of them is weakly compact must be weakly compact; consequently, if T ∈ L(E,F) admits a factorization through a reflexive Banach space, then T must be weakly compact. The converse is true as shown by the following important result.

19.12 <target id="page_260" target-type="page">260</target>Theorem (Davis/Figiel/Johnson/Pelczynski: [<xref ref-type="bibr" rid="ref22">1974]</xref>)

Let E and F be Banach spaces and T ∈ L(E,F). Then T is weakly compact if and only if T admits a factorization through a reflexive Banach space; in other words, there exists a reflexive Banach space G and operators S ∈ L(G,F) and R ∈ L(E,G) such that T = SR.

For a proof, see Davis/Figiel/Johnson/Pelczynski [1974]) or Pietsch [1980, pp.55–57].

19.f Grothendieck’s characterization for locally weakly compact operators (see Köthe [<xref ref-type="bibr" rid="ref65">1979</xref>, p.204]):

Let X and Y be LCS and T ∈ L(X, Y). Then the following two statements are equivalent.

T is locally weakly compact.

T''X'' ⊂ K Y ( Y ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3280.tif"/> .

Moreover, (i) (or (ii)) implies the following;

(iii) T^' sends equicontinuous subsets of Y^' into relatively σ ( X ′ ,X'' ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3281.tif"/> –compact subsets of X^'.

If Y is complete, then (iii) implies (i).

(It should be noted that (19.f) is actually a generalization of (19.11).)

19.g A characterization for completely continuous operators (see Pietsch [<xref ref-type="bibr" rid="ref86">1980</xref>, p.61]):

Let E and F be Banach spaces and T ∈ L(E,F). Then T is completely continuous if and only if for any Banach space E₀ and any weakly compact operator R : E₀ → E the operator TR : E₀ → F is compact.

19.h

Criteria for compactness, weak compactness and complete continuity of operators with ℓ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3282.tif"/> or c₀ or ℓ ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3283.tif"/> as domains: In the following, F is always assumed to be a Banach space.

(A) Let T ∈ L( ℓ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3284.tif"/> ,F) and y_n = Te_n for all n ≥ 1. (Hence {y_n} is a bounded sequence in F such that T [ ζ n ] = ∑ n=1 ∞ ζ n y n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3285.tif"/> (for all [ζ_n] ∈ ℓ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3286.tif"/> ) and | | T | | = sup n   | | y n | | https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3287.tif"/> [see (3.j) (i)′]. Consider the following statements.

261(i) {y_n} is a σ(F, F^')–null sequence.

(ii) T : ( ℓ 1 , σ ( ℓ 1 , c 0 ) ) → ( F , σ ( F ,  F ' ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3288.tif"/> is continuous.

(iii) T is weakly compact.

Then (i) ⇔ (ii) ⇒ (iii).

(The implication (i) ⇒ (ii) is essentially contained in Pietsch [1980, (3.2.3)].)

(B) Let T ∈ L(c₀,F) and y_n = Te_n for all n ≥ 1. [Hence {y_n} is a weakly summable sequence in F such that T ( [ ζ n ] ) = ∑ n ζ n y n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3289.tif"/> and | | T | | = sup   { ∑ n | < y n , g > |  : g ∈ U F ′ } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3290.tif"/> by (3.j)(ii)′.] Then the following statements are equivalent.

(i) {y_n} is summable in Y.

(ii) T is weakly compact.

(iii) T is compact.

(iv) T is completely continuous.

(The equivalence of (i) and (iii) is due to Randtke [1973] (also see Dazord [1976, P.109])).

(C) (Traves [1967, pp.454–457]). Let T ∈ L( ℓ ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3291.tif"/> ,F) and y_n = Te_n for all n ≥ 1.

Then the following statements are equivalent.

{y_n} is a summable sequence in F.

T : ( ℓ ∞ , σ ( ℓ ∞ , ℓ 1 ) ) → ( F , σ ( F ,   F ′ ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3292.tif"/> is continuous.

T ' : ( F ' , σ ( F ' , F ) ) → ( ℓ 1 , σ ( ℓ 1 ,   ℓ ∞ ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3293.tif"/> is continuous.

T is compact.

19.j

Criteria for compactness, weak compactness and complete continuity of operators with ranges contained in ℓ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3294.tif"/> or c₀ or ℓ ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3295.tif"/> : In the following, E will be always assumed to be a Banach space.

(A) (Randtke [1974]).Let S ∈ L ( E , ℓ 1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3296.tif"/> and f_n = S′en (for all n ≥ 1).[Hence {f_n} is weak* summable in E^' for which Sx = ∑ n < x , f n > e n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3297.tif"/> and | | S | | = sup   { ∑ n | < x , f n > |  : x ∈ U E } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3298.tif"/> (see (3.j) (i))]. Then the following statements are equivalent:

262(a) {f_n} is summable in E′.

(b) S is weakly compact.

(c) S is compact.

(B) (Pietsch [1980,p.61] and Dazord [1976,p.108]).Let S ∈ L(E,c₀) and f_n = S′e_n (for all n ≥ 1). [Hence {f_n} is a σ(E^',E)–null sequence in E^' such that S ( x ) = ∑ n < x , f n > e n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3299.tif"/> and | | S | | = sup n   | | f n | | https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3300.tif"/> (see (3.j)(ii)).] Consider the following statements:

(Wi) {f_n} is a σ ( E ′ ,E'' ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3301.tif"/> –null sequence in E^'.

(Wii) S is weakly compact.

(Ci) {f_n} is a null sequence in the Banach dual E^'.

(Cii) S is compact.

Then (Wi) ⇔ (Wii) and (Ci) ⇔ (Cii).

As an application of (19.j)(B) and (19.k)(B), we obtain immediately (19.6).