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Abstract:
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The subject of this dissertation is the study of the belonging of weak operator in-
tegrals in appropriate ideals of compact operators, as well as the investigation of perturbation
inequalities. These questions were previously considered in [16], where Cauchy–Schwarz type
inequalities were established. In addition to providing norm estimates, these inequalities also
yield sufficient conditions for an operator integral to belong to a given ideal. In the first part
of the dissertation, using these inequalities, perturbation norm inequalities are derived for
elementary operators generated by analytic functions. Specially, for an analytic function f,
trigonometric polynomials T, S : R → C and t ∈ R, if fT S,t, f¯T T,t and f¯SS,t are the associated
analytic functions, and if X ∈ B(H) and the operator P∞
n=1(AnXBn − CnXDn) belongs to a
symmetric norming (s.n.) ideal CΦ(H), for some s.n. function Φ, then the following inequality
holds
∞X
n=1
(A∗
nAn− C∗
nCn)
1
2
fT S,t
∞X
n=1
An⊗Bn
X − fT S,t
∞X
n=1
Cn⊗Dn
X
∞X
n=1
(BnB∗
n − DnD∗
n)
1
2
Φ
⩽ f¯T T,t
∞X
n=1
A∗
nAn
− f¯T T,t
∞X
n=1
C∗
nCn
1
2 ∞X
n=1
(AnXBn − CnXDn)
× f¯SS,t
∞X
n=1
BnB∗
n
− f¯SS,t
∞X
n=1
DnD∗
n
1
2
Φ
,
under certain conditions on the families (An)∞
n=1, (Bn)∞
n=1, (Cn)∞
n=1 and (Dn)∞
n=1 in B(H).
Next, the dissertation considers vector measures induced by weak∗ integrable operator-
valued functions taking values in Shatten–von Neumann ideals. Furthermore, the criteria
for the compactness and nuclearity of the Gel’fand integral are derived, with emphasis on
positive operator-valued functions.
Finally, depending on the properties of the symmetric norming function Φ, the conse-
quences of the condition
sup
e,f ∈B
Z
Ω
Φ((⟨Aten, fn⟩)∞
n=1)dμ(t) < +∞.
are explored. More precisely, it is proved that the weak∗ integral belongs to the symmetric
ideal CΦ(H), as well as the Gelfand and Pettis integrability of the CΦ(H)-valued function A . |