Reducing subspaces of multiplication operators on function spaces

This survey presents the brief history and recent development on commutants and reducing subspaces of multiplication operators on both the Hardy space and the Bergman space, and von Neumann algebras generated by multiplication operators on the Bergman space.

Reducing Subspaces of Toeplitz Operators on N_φ-type Quotient Modules on the Torus

In this paper, we prove that the Toeplitz operator with finite Blaschke product symbol Sψ（z） on Nφ has at least m non-trivial minimal reducing subspaces, where m is the dimension of H^2（Гω）⊙φ（ω）H^2（Гω）. Moreover, the restriction of Sψ（z） on any of these minimal reducing subspaces is unitary equivalent to the Bergman shift Mz.

Reducing Subspaces of Toeplitz Operators Induced by a Class of Non-analytic Monomials over the Unit Ball

In this paper,we describe the minimal reducing subspaces of Toeplitz operators induced by non-analytic monomials on the weighted Bergman spaces and Dirichlet spaces over the unit ball B_(2).It is proved that each minimal reducing subspace M is finite dimensional,and if dim M≥3,then M is induced by a monomial.Furthermore,the structure of commutant algebra v(T_(w)N_(z)N):={M^(*)_(w)NM_(z)N,M^(*)_(z)NM_(w)N}′is determined by N and the two dimensional minimal reducing subspaces of(T_(w)N_(z)N.We also give some interesting examples.

Reducing subspaces of multiplication operators with the symbol αz^k+ βw^l on L_a^2(D^2)

Multiplication operators defined on function spaces have been receiving enormous attention from both operator-theoretic and function-theoretic experts. One of the problems is to study reducing subspaces of them. The one-variable case has obtained fruitful remarkable results. However, little has been done in the multi-variable case. Under the setting of the Bergman space L2a(D2), this paper addresses those multiplication operators Mp defined by special polynomials p, where p(z, w) = αzk+ βwl, α, β∈ C. Those reducing subspaces of Mp are completely determined.

Reducing subspaces of tensor products of weighted shifts

A unilateral weighted shift A is said to be simple if its weight sequence {α_n} satisfies ▽~3(α_n^2)≠0for all n≥2.We prove that if A and B are two simple unilateral weighted shifts,then AI+IB is reducible if and only if A and B are unitarily equivalent.We also study the reducing subspaces of A^kI+IB^l and give some examples.As an application,we study the reducing subspaces of multiplication operators Mzk+αωl on function spaces.

Generalized multiresolution structures in reducing subspaces of L^2(R^d)

In this paper, we introduce the notion of generalized multiresolution structure (GMS) in the set-ting of reducing subspaces of L2(Rd). For a general expansive matrix, we obtain a necessary and sufficient condition for GMS, and prove the existence of GMS in a reducing subspace. Using GMS, we obtain a pyramid decomposition and a frame-like expansion for signals in reducing subspaces.

Generalized Multiresolution Structures in Reducing Subspaces of Local Fields

In this article,we introduce the notion of general multiresolution structure(GMS)in the reducing subspace over local fields.We show that the GMS is admitted by an arbitrary reducing subspace and characterize all those GMSs which admit a pyramids decomposition.Towards the culmination,we obtain a frame-like expansion for signals in reducing subspaces in terms of GMS over local fields.

On a Class of Weak Nonhomogeneous Affine Bi-frames for Reducing Subspaces of L^2(R^d)

For refinable functiombased affine bi-frames, nonhomogeneous ones admit fast algorithms and have extension principles as homogeneous ones. But all extension principles are based on some restrictions on refinable functions. So it is natural to ask what are expected from general refinable functions. In this paper, we introduce the notion of weak nonhomogeneous affine bi-frame （WNABF）. Under the setting of reducing subspaces of L2（Rd）, we characterize WNABFs and obtain a mixed oblique extension principle for WNABFs based on general refinable functions.

REDUCIBILITY FOR A CLASS OF ANALYTIC MULTIPLIERS ON SOBOLEV DISK ALGEBRA

We prove the reducibility of analytic multipliers M_(φ)with a class of finite Blaschke products symbolφon the Sobolev disk algebra R(D).We also describe their nontrivial minimal reducing subspaces.