Volume 6, Issue 1

We study the truncated multidimensional moment problem with a general type of truncations. The operator approach to the moment problem is presented. The case where the associated operators form a commuting self-adjoint tuple is characterized in terms of the given moments. The case of the dimensional stability is characterized in terms of the prescribed moments as well. Some sufficient conditions for the solvability of the moment problem are presented. A construction of the corresponding solution is described by algorithms. Numerical examples of the construction are provided.

We consider the Dirichlet problem on in_nite and locally _nite rooted trees, andwe prove that the set of irregular points for continuous data has zero capacity. We also give some uniqueness results for solutions in Sobolev W 1, p of the tree.

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 à ( 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)$${\bf{ber}}(AB){\bf{ber}}(A){\bf{ber}}(B)$$

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.

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.

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.

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 BMO 1 symbols, and with Berezin transforms of vanishing oscillation.

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.

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 T n x 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 ( n k ) 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.