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Abstract:
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In the rst and the second chapter of dissertation we prove
some new inequalities for the spectral radius, essential spectral radius, oper-
ator norm, measure of non-compactness and numerical radius of Hadamard
(Schur) weighted geometric means of positive kernel operators on Banach
function and sequence spaces. The list of extensions and re nings of known
inequalities has been expanded. Some new inequalities and equalities for
the generalized and the joint spectral radius and their essential versions of
Hadamard (Schur) geometric means of bounded sets of positive kernel op-
erators on Banach function spaces have been proved. There are additional
results in case of non-negative matrices that de ne operators on Banach
sequence spaces. In the third part we present some inequalities for opera-
tor monotone functions and (co)hyponormal operators and give relations of
Schur multipliers to derivation like inequalities for operators. In particular,
let Ψ, Φ be s.n. functions, p ⩾ 2 and φ be an operator monotone function on
[0, ∞) such that φ(0) = 0. If A, B, X ∈ B(H) and A and B are strictly ac-
cretive such that AX−XB ∈ CΨ(H), then also AXφ(B)−φ(A)XB ∈ CΨ(H)
and
||AXφ(B) − φ(A)XB||Ψ ⩽
r
φ
A+A∗
2
− A+A∗
2 φ′
A+A∗
2
A+A∗
2
−1
A(AX − XB)B
B+B∗
2
−1
r
φ
B+B∗
2
− B+B∗
2 φ′
B+B∗
2
Ψ
.
under any of the following conditions:
(a) Both A and B are normal,
(b) A is cohyponormal, B is hyponormal and at least one of them is normal,
and Ψ := Φ(p)∗
,
(c) A is cohyponormal, B is hyponormal and ||.||Ψ is the trace norm ||.||1.
Alternative inequalities for ||.||Ψ(p) norms are also obtained. |