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Abstract:
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This doctoral dissertation addresses the integration of functions taking values in
spaces of bounded operators and in spaces of complex measures on a given σ-algebra. The
mentioned integrability is considered in a more general sense than that required in the theory
of weak integration of vector-valued functions. The first part of the dissertation deals with
the integrability of families of operators. If (Ω, M, μ) is a space with a positive measure μ
and (At)t∈Ω is a family of operators from B(X, Y ), where X and Y are Banach spaces, then
μ-integrability of the function Ω ∋ t 7 → ⟨Atx, y∗⟩ ∈ C is required for every x ∈ X and y∗ ∈ Y ∗.
In this case, we prove that the quantity sup∥x∥=∥y∗∥=1
R
Ω ⟨Atx, y∗⟩ dμ(t) is finite. This expres-
sion allows us to define a norm on the corresponding vector space of families of operators.
Furthermore, for every E ∈ M, one obtains an operator R
E At dμ(t) in B(X, Y ∗∗), whose
defining property is ⟨y∗, R
E At dμ(t) x⟩ = R
E ⟨Atx, y∗⟩ dμ(t) for every x ∈ X and y∗ ∈ Y ∗. The
second part of the dissertation deals with the integrability of families of measures. If (λx)x∈X
is a family of complex measures on (Y, A), where (X, B, μ) is a space with a positive measure
μ, and if for every A ∈ A the function X ∋ x 7 → λx(A) ∈ C is μ-integrable, then the quantity
supA∈A
R
X |λx(A)| dμ(x) is finite. This allows us to define a norm on the corresponding vector
space of families of measures. In this case, for every B ∈ B there exists a complex measure
R
B λx dμ(x) on A such that
R
B λx dμ(x)
(A) = R
B λx(A) dμ(x) for every A ∈ A. The dis-
sertation is organized as follows. The first part (Chapters 2–4) deals with the integration of
functions taking values in B(X, Y ). Chapter 2 provides a survey of the known results on the
integration of functions in B(H), where H is a separable Hilbert space, and presents original
results extending the existing theory. In Chapter 3, the developed theory is applied to the
Laplace transform of B(H)-valued functions, which has been previously considered in the
literature. Chapter 4 is significant because it generalizes the integrability of functions taking
values in B(X, Y ). This type of integration was first defined in [8]. The second part of the
dissertation (Chapter 5) deals with the integration of functions taking values in spaces of
complex measures on a given σ-algebra. The introduced type of integration is more general
than Pettis concept and has been considered in [6, 7]. These works represent a natural ex-
tension and application of the experiences gained from working with functions taking values
in operator spaces, including original results of the candidate with coauthors. Numerous
concrete examples are included, making this abstract material much more illustrative. |