## Annals of Functional Analysis

### Minimal reducing subspaces of an operator-weighted shift

#### Abstract

We introduce a family $\mathcal{T}$ consisting of invertible matrices with exactly one nonzero entry in each row and each column. The elements of $\mathcal{T}$ are possibly mutually noncommuting, and they need not be normal or self-adjoint. We consider an operator-valued unilateral weighted shift $W$ with a uniformly bounded sequence of weights belonging to $\mathcal{T}$, and we describe its minimal reducing subspaces.

#### Article information

Source
Ann. Funct. Anal. Volume 8, Number 4 (2017), 531-546.

Dates
Accepted: 9 January 2017
First available in Project Euclid: 29 June 2017

https://projecteuclid.org/euclid.afa/1498723220

Digital Object Identifier
doi:10.1215/20088752-2017-0017

#### Citation

Hazarika, Munmun; Gogoi, Pearl S. Minimal reducing subspaces of an operator-weighted shift. Ann. Funct. Anal. 8 (2017), no. 4, 531--546. doi:10.1215/20088752-2017-0017. https://projecteuclid.org/euclid.afa/1498723220

#### References

• [1] A. Bourhim, Spectrum of bilateral shifts with operator-valued weights, Proc. Amer. Math. Soc. 134 (2006), no. 7, 2131–2137.
• [2] T. J. Ferguson, Extremal problems in Bergman spaces and an extension of Ryabykh’s theorem, Ill. J. Math. 55 (2011), no. 2, 555-573.
• [3] R. Gellar, Operators commuting with a weighted shift, Proc. Amer. Math. Soc. 23 (1969), 538–545.
• [4] J. Guyker, On reducing subspaces of normally weighted bilateral shifts, Houston J. Math. 11 (1985), no. 4, 515–521.
• [5] M. Hazarika and S. C. Arora, Minimal reducing subspaces of the unilateral shift operator on an operator weighted sequence space, Indian J. Pure Appl. Math. 35 (2004), no. 6, 747–757.
• [6] D. Herrero, Spectral pictures of hyponormal bilateral operator weighted shifts, Proc. Amer. Math. Soc. 109 (1990), no. 3, 753–763.
• [7] J. Hu, S. Sun, X. Xu, and D. Yu, Reducing subspace of analytic Toeplitz operators on the Bergman space, Integral Equations Operator Theory 49 (2004), no. 3, 387–395.
• [8] Z. Jabloński, Hyperexpansive operator valued unilateral weighted shifts, Glasg. Math. J. 46 (2004), no. 2, 405–416.
• [9] A. Lambert, Unitary equivalence and reducibility of invertibly weighted shifts, Bull. Aust. Math. Soc. 5 (1971), 157–173.
• [10] J. X. Li, Y. Q. Ji, and S. L. Sun, The essential spectrum and Banach reducibility of operator weighted shifts, Acta Math. Sinica Engl. Ser. 17 (2001), no. 3, 413–424.
• [11] N. K. Nikol’skii, Invariant subspaces of weighted shift operators, Mat. Sb. (N.S.) 74 (116) 1967, 172–190.
• [12] C. M. Pearcy and S. Petrovic, On polynomially bounded weighted shifts, Houston J. Math. 20 (1994), no. 1, 27–45.
• [13] V. S. Pilidi, On invariant subspaces of multiple weighted shift operators, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 2, 373–398, 480.
• [14] A. L. Shields, “Weighted shift operators and analytic function theory,” in Topics in Operator Theory, Math. Surveys Monogr. 13, Amer. Math. Soc., Providence, 1974, 49-128.
• [15] M. Stessin and K. Zhu, Reducing subspaces of weighted shift operators, Proc. Amer. Math. Soc. 130 (2002), no. 9, 2631–2639.
• [16] K. Zhu, Reducing subspaces for a class of multiplication operators, J. London Math. Soc. 62 (2000), no. 2, 553–568.