TY - JOUR AU - Gupta, Rajeev AU - Kumar, Surjit AU - Trivedi, Shailesh TI - Unitary equivalence of operator-valued multishifts JF - JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS J2 - J MATH ANAL APPL VL - 487 PY - 2020 IS - 2 PG - 23 SN - 0022-247X DO - 10.1016/j.jmaa.2020.124032 UR - https://m2.mtmt.hu/api/publication/31492847 ID - 31492847 AB - We systematically study various aspects of operator-valued multishifts. Beginning with basic properties, we show that the class of multishifts on the directed Cartesian product of rooted directed trees is contained in that of operator-valued multishifts. Further, we establish circularity, analyticity and wandering subspace property of these multishifts. In the rest part of the paper, we study the function theoretic behaviour of operator-valued multishifts. We determine the bounded point evaluation, reproducing kernel structure and the unitary equivalence of operator-valued multishifts with invertible operator weights. In contrast with a result of Lubin, it appears that the set of all bounded point evaluations of an operator-valued multishift may be properly contained in the joint point spectrum of the adjoint of underlying multishift. (C) 2020 Elsevier Inc. All rights reserved. LA - English DB - MTMT ER - TY - JOUR AU - Pietrzycki, Pawel TI - Generalized Multipliers for Left-Invertible Operators and Applications JF - INTEGRAL EQUATIONS AND OPERATOR THEORY J2 - INTEGR EQUAT OPER TH VL - 92 PY - 2020 IS - 5 PG - 31 SN - 0378-620X DO - 10.1007/s00020-020-02598-1 UR - https://m2.mtmt.hu/api/publication/31730790 ID - 31730790 AB - Generalized multipliers for a left-invertible operator T, whose formal Laurent series U-x(z) = Sigma(infinity)(n=1)(PETn x)1/z(n) + Sigma(infinity)(n=0) (PET/*n x)z(n), x is an element of H actually represent analytic functions on an annulus or a disc are investigated. We show that they are coefficients of analytic functions and characterize the commutant of some left-invertible operators, which satisfies certain conditions in its terms. In addition, we prove that the set of multiplication operators associated with a weighted shift on a rootless directed tree lies in the closure of polynomials in z and 1/z of the weighted shift in the topologies of strong and weak operator convergence. LA - English DB - MTMT ER - TY - JOUR AU - Anand, Akash AU - Chavan, Sameer AU - Jablonski, Zenon Jan AU - Stochel, Jan TI - A solution to the Cauchy dual subnormality problem for 2-isometries JF - JOURNAL OF FUNCTIONAL ANALYSIS J2 - J FUNCT ANAL VL - 277 PY - 2019 IS - 12 PG - 51 SN - 0022-1236 DO - 10.1016/j.jfa.2019.108292 UR - https://m2.mtmt.hu/api/publication/30989208 ID - 30989208 AB - The Cauchy dual subnormality problem asks whether the Cauchy dual operator T' := T(T*T)(-1) of a 2-isometry T is subnormal. In the present paper we show that the problem has a negative solution. The first counterexample depends heavily on a reconstruction theorem stating that if T is a 2-isometric weighted shift on a rooted directed tree with nonzero weights that satisfies the perturbed kernel condition, then T' is subnormal if and only if T satisfies the (unperturbed) kernel condition. The second counterexample arises from a 2-isometric adjacency operator of a locally finite rooted directed tree again by thorough investigations of positive solutions of the Cauchy dual subnormality problem in this context. We prove that if T is a 2-isometry satisfying the kernel condition or a quasi-Brownian isometry, then T' is subnormal. We construct a 2-isometric adjacency operator T of a rooted directed tree such that T does not satisfy the kernel condition, T is not a quasi-Brownian isometry and T' is subnormal. (C) 2019 Elsevier Inc. All rights reserved. LA - English DB - MTMT ER - TY - JOUR AU - Budzynski, P. AU - Dymek, P. AU - Planeta, A. AU - Ptak, M. TI - Weighted shifts on directed trees: their multiplier algebras, reflexivity and decompositions JF - STUDIA MATHEMATICA J2 - STUD MATH VL - 244 PY - 2019 IS - 3 SP - 285 EP - 308 PG - 24 SN - 0039-3223 DO - 10.4064/sm170220-20-9 UR - https://m2.mtmt.hu/api/publication/30546604 ID - 30546604 AB - We study bounded weighted shifts on directed trees. We show that the set of multiplication operators associated with an injective weighted shift on a rooted directed tree coincides with the WOT/SOT closure of the set of polynomials of the weighted shift. From this fact we deduce reflexivity of those weighted shifts on rooted directed trees whose all path-induced spectral-like radii are positive. We show that weighted shifts with positive weights on rooted directed trees admit a Wold-type decomposition. We prove that the pairwise orthogonality of the factors in the decomposition is equivalent to the weighted shift being balanced. LA - English DB - MTMT ER - TY - JOUR AU - Pietrzycki, Pawel TI - A Shimorin-type analytic model on an annulus for left-invertible operators and applications JF - JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS J2 - J MATH ANAL APPL VL - 477 PY - 2019 IS - 2 SP - 885 EP - 911 PG - 27 SN - 0022-247X DO - 10.1016/j.jmaa.2019.04.027 UR - https://m2.mtmt.hu/api/publication/30989209 ID - 30989209 AB - A new analytic model for left-invertible operators, which extends both Shimorin's analytic model for left-invertible and analytic operators and Gellar's model for bilateral weighted shift is introduced and investigated. We show that a left-invertible operator T, which satisfies certain conditions can be modeled as a multiplication operator M-z on a reproducing kernel Hilbert space of vector-valued analytic functions on an annulus or a disc. A similar result for composition operators in l(2)-spaces is established. (C) 2019 Elsevier Inc. All rights reserved. LA - English DB - MTMT ER - TY - CHAP AU - Budzynski, Piotr AU - Jablonski, Zenon AU - Jung, Il Bong AU - Stochel, Jan TI - Unbounded Weighted Composition Operators in L-2-Spaces Preface T2 - UNBOUNDED WEIGHTED COMPOSITION OPERATORS IN L2-SPACES PB - Springer Netherlands CY - Cham (Németország) T3 - Lecture Notes in Mathematics, ISSN 0075-8434 PY - 2018 SP - VII EP - + PG - 8 UR - https://m2.mtmt.hu/api/publication/30546605 ID - 30546605 LA - English DB - MTMT ER - TY - JOUR AU - Pietrzycki, Pawel TI - Reduced commutativity of moduli of operators JF - LINEAR ALGEBRA AND ITS APPLICATIONS J2 - LINEAR ALGEBRA APPL VL - 557 PY - 2018 SP - 375 EP - 402 PG - 28 SN - 0024-3795 DO - 10.1016/j.laa.2018.08.007 UR - https://m2.mtmt.hu/api/publication/30464593 ID - 30464593 AB - In this paper, we investigate the question of when the equations A*(s)A(s) = (A*A)(s), s is an element of S, where S is a finite set of positive integers, imply the quasinormality or normality of A. In particular, it is proved that if S = {p, m, m + p, n, n +p}, where p >= 1 and 2 <= m < n, then A is quasinormal. Moreover, if A is invertible and S = {m, n, n+m} with m <= n, then A is normal. The case when S = {m, m+n} and A*(n)A(n) <= (A*A)(n) is also discussed. (C) 2018 Elsevier Inc. All rights reserved. LA - English DB - MTMT ER -