ABSTRACT
Let, for j = 0, 1, H j = H j * ≥ − ω j > − ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/inq_chapter38_453_1.tif"/> , be a self-adjoint operator in the Hilbert spaces ℋ j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/inq_chapter38_453_2.tif"/> . Let T : ℋ 1 → ℋ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/inq_chapter38_453_3.tif"/> be a linear operator with domain in ℋ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/inq_chapter38_453_4.tif"/> and range in ℋ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/inq_chapter38_453_5.tif"/> . Let Vj (t) = exp(−tHj ), t ≥ 0, be the strongly continuous semigroup generated by Hj, j = 0, 1. If the operators
( a I + H 0 ) V 0 ( t 0 ) T ( a I + H 1 ) − 1 and ( a I + H 0 ) − 1 T ( a I + H 1 ) V 1 ( t 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/equ_chapter38_453_1.tif"/>are compact, (Hilbert-Schmidt, Trace class), then so is the operator
∫ 0 t 0 V 0 ( u ) T V 1 ( t 0 − u ) d u . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/equ_chapter38_453_2.tif"/>The result is applicable if T = J H 1 − H 0 J https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/inq_chapter38_453_6.tif"/> , where J : ℋ 1 → ℋ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/inq_chapter38_453_7.tif"/> is a bounded linear (identification) operator. In this case ∫ 0 t 0 V 0 ( u ) T V 1 ( t 0 − u ) d u = V 0 ( t 0 ) J − J V 1 ( t 0 ) ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/inq_chapter38_453_8.tif"/>
i.e. the difference of the semigroups. Some convergence and approximation results are presented as well. For example the operator ∫ 0 t 0 V 0 ( u ) T V 1 ( t 0 − u ) d u https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/inq_chapter38_453_9.tif"/> is expressed in terms of the operator t 0 V 0 (t 0/2)TV 1 (t 0/2).
