Unitarily equivalent bilateral weighted shifts with operator weights

• Autorzy: Buchała, Michał
• Języki publikacji: EN

Abstrakty

EN

The aim of this paper is to study unitarily equivalent bilateral weighted shifts with operator weights. Our purpose is to establish a general characterization of unitary equivalence of such shifts under the assumption that the weights are quasi-invertible. Under further assumptions on weights it was proved that unitary equivalence of bilateral weigthed shifts with operator weights defined on C2 can always be given by a unitary operator with at most two non-zero diagonals. The paper contains also examples of unitarily equivalent shifts with weights defined on Ck such that every unitary operator, which intertwines them has at least k non-zero diagonals.

Słowa kluczowe

EN

weighted shifts; operator weights; unitary equivalence

• Czasopismo: Opuscula Mathematica
• Rocznik: 2024
• Tom: Vol. 44, no. 5
• Strony: 651--672
• Opis fizyczny: Bibliogr. 16 poz., tab.

Twórcy

autor

Buchała, Michał

• Doctoral School of Exact and Natural Sciences, Jagiellonian University, Łojasiewicza 11, PL-30348 Kraków, Poland,
• Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, PL-30348 Kraków, Poland

Bibliografia

• [1] A. Anand, S. Chavan, Z.J. Jabłoński, J. Stochel, Complete systems of unitary invariants for some classes of 2-isometries, Banach J. Math. Anal. 13 (2019), no. 2, 359–385.
• [2] S.K. Berberian, Note on a theorem of Fuglede and Putnam, Proc. Amer. Math. Soc. 10 (1959), 175–182.
• [3] J.B. Conway, A Course in Functional Analysis, Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1985.
• [4] J. Guyker, On reducing subspaces of normally weighted bilateral shifts, Houston J. Math. 11 (1985), no. 4, 515–521.
• [5] P.R. Halmos, A Hilbert Space Problem Book, Graduate Texts in Mathematics, vol. 19, Springer-Verlag, New York–Berlin, 2nd edition, 1982.
• [6] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
• [7] N. Ivanovski, Similarity and quasisimilarity of bilateral operator valued weighted shifts, Mat. Bilten 43 (1993) 17, 33–37.
• [8] N. Ivanovski, M. Orovčanec, On similarity and quasisimilarity of unilateral operator valued weighted shifts, Mat. Bilten 35/36 (1985/86), no. 9–10, 5–10.
• [9] J. Kośmider, On unitary equivalence of bilateral operator valued weighted shifts, Opuscula Math. 39 (2019), no. 4, 543–555.
• [10] A. Lambert, Unitary equivalence and reducibility of invertibly weighted shifts, Bull. Austral. Math. Soc. 5 (1971), 157–173.
• [11] M. Maiuriello, Dynamics of Linear Operators, Aracne (Genzano di Roma), 2022.
• [12] M. Orovčanec, Unitary equivalence of unilateral operator valued weighted shifts with quasi-invertible weights, Mat. Bilten 43 (1993) 17, 45–50.
• [13] V.S. Pilidi, Invariant subspaces of multiple weighted shift operators, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 2, 373–398.
• [14] F. Riesz, B. Sz.-Nagy, Functional Analysis, Dover Books on Advanced Mathematics, Dover Publications, Inc., New York, French edition, 1990.
• [15] A.L. Shields, Weighted shift operators and analytic function theory, [in:] Topics in Operator Theory, Math. Surveys, vol. 13, Amer. Math. Soc., Providence, R.I., 1974, 49–128.
• [16] J. Weidmann, Linear Operators in Hilbert Spaces, Graduate Texts in Mathematics, vol. 68, Springer-Verlag, New York–Berlin, 1980.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023)

Identyfikatory

DOI

10.7494/OpMath.2024.44.5.651