Nonnormal Classes of Operators

If N is a normal operator in L(H), the set of all bounded linear operators on a Hilbert space H, then it is known that it has a number of very interesting properties; among them we note the following:

rN = ‖N‖. 2. conv https://www.w3.org/1998/Math/MathML"> σ ( N ) ⁢       = ⁢         W ( ⁢   T ) ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003065050/785283b7-c566-4cdf-8808-2e10713aead5/content/inline-math257.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .

‖Tx‖ = ‖T*x‖ for all × ϵ H.

For all complex numbers z, N + zI = N + z is a normal operator.

If H_l is an invariant subspace of N, N/H_l has property l.

If H_l is an invariant subspace for N and N*; then N/H_l is normal.

σ(T) and https://www.w3.org/1998/Math/MathML"> W ( ⁢   T ) ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003065050/785283b7-c566-4cdf-8808-2e10713aead5/content/inline-math258.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> are spectral sets of T. ^†

r_N = w(T) .

N has the G_l property.

Re σ (T) = σ (Re T) .