Unitarily equivalent bilateral weighted shifts with operator weights
Abstract: We study unitarily equivalent bilateral weighted shifts with operator weights. We establish a general characterization of unitary equivalence of such shifts under the assumption that the weights are quasi-invertible. We prove that under certain assumptions unitary equivalence of bilateral weighted shifts with operator weights defined on $ \mathbb{C}{2} $ can always be given by a unitary operator with at most two non-zero diagonals. We provide examples of unitarily equivalent shifts with weights defined on $ \mathbb{C}{k} $ such that every unitary operator, which intertwines them has at least $ k $ non-zero diagonals.
- Complete systems of unitary invariants for some classes of 2-isometries. Banach J. Math. Anal., 13(2):359–385, 2019.
- S. K. Berberian. Note on a theorem of Fuglede and Putnam. Proc. Amer. Math. Soc., 10:175–182, 1959.
- John B. Conway. A course in functional analysis, volume 96 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1985.
- James Guyker. On reducing subspaces of normally weighted bilateral shifts. Houston J. Math., 11(4):515–521, 1985.
- Paul Richard Halmos. A Hilbert space problem book, volume 17 of Encyclopedia of Mathematics and its Applications. Springer-Verlag, New York-Berlin, second edition, 1982. Graduate Texts in Mathematics, 19.
- Matrix analysis. Cambridge University Press, Cambridge, 1985.
- Novak Ivanovski. Similarity and quasisimilarity of bilateral operator valued weighted shifts. Mat. Bilten, 43(17):33–37, 1993.
- On similarity and quasisimilarity of unilateral operator valued weighted shifts. Mat. Bilten, 35/36(9-10):5–10 (1989), 1985/86.
- Jakub Kośmider. On unitary equivalence of bilateral operator valued weighted shifts. Opuscula Math., 39(4):543–555, 2019.
- Alan Lambert. Unitary equivalence and reducibility of invertibly weighted shifts. Bull. Austral. Math. Soc., 5:157–173, 1971.
- Marija Orovčanec. Unitary equivalence of unilateral operator valued weighted shifts with quasi-invertible weights. Mat. Bilten, 43(17):45–50, 1993.
- V. S. Pilidi. Invariant subspaces of multiple weighted shift operators. Izv. Akad. Nauk SSSR Ser. Mat., 43(2):373–398, 480, 1979.
- Functional analysis. Dover Books on Advanced Mathematics. Dover Publications, Inc., New York, french edition, 1990. Reprint of the 1955 original.
- Allen L. Shields. Weighted shift operators and analytic function theory. In Topics in operator theory, Math. Surveys, No. 13, pages 49–128. Amer. Math. Soc., Providence, R.I., 1974.
- Joachim Weidmann. Linear operators in Hilbert spaces, volume 68 of Graduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1980. Translated from the German by Joseph Szücs.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.