Journal of the Mathematical Society of Japan

Common reducing subspaces of several weighted shifts with operator weights

Caixing GU

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We characterize common reducing subspaces of several weighted shifts with operator weights. As applications, we study the common reducing subspaces of the multiplication operators by powers of coordinate functions on Hilbert spaces of holomorphic functions in several variables. The identification of reducing subspaces also leads to structure theorems for the commutants of von Neumann algebras generated by these multiplication operators. This general approach applies to weighted Hardy spaces, weighted Bergman spaces, Drury–Arveson spaces and Dirichlet spaces of the unit ball or polydisk uniformly.

Article information

J. Math. Soc. Japan, Volume 70, Number 3 (2018), 1185-1225.

Received: 16 March 2016
Revised: 31 January 2017
First available in Project Euclid: 25 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15] 47A15: Invariant subspaces [See also 47A46]
Secondary: 46E22: Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32] 32A35: Hp-spaces, Nevanlinna spaces [See also 32M15, 42B30, 43A85, 46J15] 32A36: Bergman spaces

weighted shifts with operator weights reducing subspaces analytic Toeplitz operators weighted Hardy space on unit ball weighted Bergman spaces Dirichlet space on polydisk


GU, Caixing. Common reducing subspaces of several weighted shifts with operator weights. J. Math. Soc. Japan 70 (2018), no. 3, 1185--1225. doi:10.2969/jmsj/74677467.

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