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A subnormal weighted shift on a directed tree whose nth power has trivial domain
Budzynski, P.,Jablonski, Z.J.,Jung, I.B.,Stochel, J. Academic Press 2016 Journal of mathematical analysis and applications Vol.435 No.1
<P>It is shown that for any positive integer n there exists a subnormal weighted shift on a directed tree whose nth power is closed and densely defined while its (n+1)th power has trivial domain. Similar result for composition operators in L-2-spaces is established. (C) 2015 Elsevier Inc. All rights reserved.</P>
Subnormality of unbounded composition operators over one-circuit directed graphs: Exotic examples
Budzynski, P.,Jablonski, Z.J.,Jung, I.B.,Stochel, J. Academic Press ; Elsevier Science B.V. Amsterdam 2017 Advances in mathematics Vol.310 No.-
<P>A recent example of a non-hyponormal injective composition operator in an L-2-space generating Stieltjes moment sequences, invented by three of the present authors, was built over a non-locally finite directed tree. The main goal of this paper is to solve the problem of whether there exists such an operator over a locally finite directed graph and, in the affirmative case, to find the simplest possible graph with these properties (simplicity refers to local valency). The problem is solved affirmatively for the locally finite directed graph g(2,0), which consists of two branches and one loop. The only simpler directed graph for which the problem remains unsolved consists of one branch and one loop. The consistency condition, the only efficient tool for verifying subnormality of unbounded composition operators, is intensively studied in the context of g(2),(0), which leads to a constructive method of solving the problem. The method itself is partly based on transforming the Krein and the Friedrichs measures coming either from shifted Al-Salam-Carlitz q-polynomials or from a quartic birth and death process. (C) 2017 Elsevier Inc. All rights reserved.</P>
A multiplicative property characterizes quasinormal composition operators in L<sup>2</sup>-spaces
Budzynski, P.,Jablonski, Z.J.,Jung, I.B.,Stochel, J. Academic Press 2014 Journal of mathematical analysis and applications Vol.409 No.1
A densely defined composition operator in an L<SUP>2</SUP>-space induced by a measurable transformation φ is shown to be quasinormal if and only if the Radon-Nikodym derivatives h<SUB>φ^n</SUB> attached to powers φ<SUP>n</SUP> of φ have the multiplicative property: h<SUB>φ^n</SUB>=h<SUB>φ</SUB><SUP>n</SUP> almost everywhere for n=0,1,2,....
A NOTE ON k-HYPERREFLEXIVITY OF TOEPLITZ-HARMONIC SUBSPACES
Budzynski, Piotr,Piwowarczyk, Kamila,Ptak, Marek Korean Mathematical Society 2014 대한수학회보 Vol.51 No.6
The 2-hyperreflexivity of Toeplitz-harmonic type subspace generated by an isometry or a quasinormal operator is shown. The k-hyperreflexivity of the tensor product $\mathcal{S}{\otimes}\mathcal{V}$ of a k-hyperreflexive decom-posable subspace $\mathcal{S}$ and an abelian von Neumann algebra $\mathcal{V}$ is established.
Unbounded subnormal weighted shifts on directed trees. II
Budzynski, P.,Jablonski, Z.J.,Jung, I.B.,Stochel, J. Academic Press 2013 Journal of mathematical analysis and applications Vol.398 No.2
Criteria for subnormality of unbounded injective weighted shifts on leafless directed trees with one branching vertex are proposed. The case of classical weighted shifts is discussed. The relevance of an inductive limit approach to subnormality of weighted shifts on directed trees is revealed.
A NOTE ON k-HYPERREFLEXIVITY OF TOEPLITZ-HARMONIC SUBSPACES
Piotr Budzynski,Kamila Piwowarczyk,Marek Ptak 대한수학회 2014 대한수학회보 Vol.51 No.6
The 2-hyperreflexivity of Toeplitz-harmonic type subspace generated by an isometry or a quasinormal operator is shown. The k- hyperreflexivity of the tensor product S ⓧ V of a k-hyperreflexive decom- posable subspace S and an abelian von Neumann algebra V is established.