Showing papers in "Concrete Operators in 2019"

Berezin number inequalities for operators

Abstract: Abstract The Berezin transform à of an operator A, acting on the reproducing kernel Hilbert space ℋ = ℋ (Ω) over some (non-empty) set Ω, is defined by Ã(λ) = 〉Aǩ λ, ǩ λ〈 (λ ∈ Ω), where k⌢λ=kλ‖ kλ ‖ ${\\mathord{\\buildrel{\\lower3pt\\hbox{$\\scriptscriptstyle\\frown$}}\\over k} _\\lambda } = {{{k_\\lambda }} \\over {\\left\\| {{k_\\lambda }} \\right\\|}}$ is the normalized reproducing kernel of ℋ. The Berezin number of an operator A is defined by ber(A)=supλ∈Ω| A˜(λ) |=supλ∈Ω| 〈 Ak⌢λ,k⌢λ 〉 | ${\\bf{ber}}{\\rm{(}}A) = \\mathop {\\sup }\\limits_{\\lambda \\in \\Omega } \\left| {\\tilde A(\\lambda )} \\right| = \\mathop {\\sup }\\limits_{\\lambda \\in \\Omega } \\left| {\\left\\langle {A{{\\mathord{\\buildrel{\\lower3pt\\hbox{$\\scriptscriptstyle\\frown$}}\\over k} }_\\lambda },{{\\mathord{\\buildrel{\\lower3pt\\hbox{$\\scriptscriptstyle\\frown$}}\\over k} }_\\lambda }} \\right\\rangle } \\right|$ . In this paper, we prove some Berezin number inequalities. Among other inequalities, it is shown that if A, B, X are bounded linear operators on a Hilbert space ℋ, then ber(AX±XA)⩽ber12(A*A+AA*)ber12(X*X+XX*) $${\\bf{ber}}(AX \\pm XA) \\leqslant {\\bf{be}}{{\\bf{r}}^{{1 \\over 2}}}\\left( {A*A + AA*} \\right){\\bf{be}}{{\\bf{r}}^{{1 \\over 2}}}\\left( {X*X + XX*} \\right)$$ and ber2(A*XB)⩽‖ X ‖2ber(A*A)ber(B*B). $${\\bf{be}}{{\\bf{r}}^2}({A^*}XB) \\leqslant {\\left\\| X \\right\\|^2}{\\bf{ber}}({A^*}A){\\bf{ber}}({B^*}B).$$ We also prove the multiplicative inequality ber(AB)⩽ber(A)ber(B) $${\\bf{ber}}(AB){\\bf{ber}}(A){\\bf{ber}}(B)$$

On unbounded commuting Jacobi operators and some related issues

Abstract: Abstract We consider the situations, when two unbounded operators generated by infinite Jacobi matrices, are self-adjoint and commute. It is found that if two Jacobi matrices formally commute, then two corresponding operators are either self-adjoint and commute, or admit a commuting self-adjoint extensions. In the latter case such extensions are explicitly described. Also, some necessary and sufficient conditions for self-adjointness of Jacobi operators are studied.

Moment Problems in Hereditary Function Spaces

Abstract: Abstract We introduce a concept of hereditary set of multi-indices, and consider vector spaces of functions generated by families associated to such sets of multi-indices, called hereditary function spaces. Existence and uniquenes of representing measures for some abstract truncated moment problems are investigated in this framework, by adapting the concept of idempotent and that of dimensional stability, and using some techniques involving C*-algebras and commuting self-adjoint multiplication operators.

The operator approach to the truncated multidimensional moment problem

Abstract: We study the truncated multidimensional moment problem with a general type of truncations. The operator approach to the moment problem is presented. A way to construct atomic solutions of the moment problem is indicated.

The Blum-Hanson Property

Abstract: Abstract Given a (real or complex, separable) Banach space, and a contraction T on X, we say that T has the Blum-Hanson property if whenever x, y ∈ X are such that Tnx tends weakly to y in X as n tends to infinity, the means 1N∑k=1NTnkx {1 \\over N}\\sum\\limits_{k = 1}^N {{T^{{n_k}}}x} tend to y in norm for every strictly increasing sequence (nk) k≥1 of integers. The space X itself has the Blum-Hanson property if every contraction on X has the Blum-Hanson property. We explain the ergodic-theoretic motivation for the Blum-Hanson property, prove that Hilbert spaces have the Blum-Hanson property, and then present a recent criterion of a geometric flavor, due to Lefèvre-Matheron-Primot, which allows to retrieve essentially all the known examples of spaces with the Blum-Hanson property. Lastly, following Lefèvre-Matheron, we characterize the compact metric spaces K such that the space C(K) has the Blum-Hanson property.

Somewhere Dense Orbit that is not Dense on a Complex Hilbert Space

Abstract: Abstract In this paper, we present the existence of n-tuple of operators on complex Hilbert space that has a somewhere dense orbit and is not dense. We give the solution to the question stated in [11]: “Is there n-tuple of operators on a complex Hilbert space that has a somewhere dense orbit that is not dense?” We do so by extending the results due to Feldman [11] and Leòn-Saavedra [12] to complex Hilbert space. Further illustrative examples of somewhere dense orbits are given to support the results.

Quaternionic inner and outer functions

Abstract: We study properties of inner and outer functions in the Hardy space of the quaternionic unit ball. In particular, we give sufficient conditions as well as necessary ones for functions to be inner or outer.

On the closed range problem for composition operators on the Dirichlet space

Abstract: Abstract We characterize closed range composition operators on the Dirichlet space for a particular class of composition symbols. The characterization relies on a result about Fredholm Toeplitz operators with BMO1 symbols, and with Berezin transforms of vanishing oscillation.