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On the dependence of the orthogonal projector on deformations of the scalar product

Zbigniew Pasternak-Winiarski (1998)

Studia Mathematica

We consider scalar products on a given Hilbert space parametrized by bounded positive and invertible operators defined on this space, and orthogonal projectors onto a fixed closed subspace of the initial Hilbert space corresponding to these scalar products. We show that the projector is an analytic function of the scalar product, we give the explicit formula for its Taylor expansion, and we prove some algebraic formulas for projectors.

On the Taylor functional calculus

V. Müller (2002)

Studia Mathematica

We give a Martinelli-Vasilescu type formula for the Taylor functional calculus and a simple proof of its basic properties.

Operator theoretic properties of semigroups in terms of their generators

S. Blunck, L. Weis (2001)

Studia Mathematica

Let ( T t ) be a C₀ semigroup with generator A on a Banach space X and let be an operator ideal, e.g. the class of compact, Hilbert-Schmidt or trace class operators. We show that the resolvent R(λ,A) of A belongs to if and only if the integrated semigroup S t : = ∫ 0 t T s d s belongs to . For analytic semigroups, S t ∈ implies T t ∈ , and in this case we give precise estimates for the growth of the -norm of T t (e.g. the trace of T t ) in terms of the resolvent growth and the imbedding D(A) ↪ X.

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