Reducing Subspaces Research Articles

Reducing Subspaces of Multiplication Operators on the Dirichlet Space

In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product $${\phi}$$ on the Dirichlet space D. We prove that any two distinct nontrivial minimal reducing subspaces of $${M_\phi}$$ are orthogonal. When the order n of $${\phi}$$ is 2 or 3, we show that $${M_\phi}$$ is reducible on D if and only if $${\phi}$$ is equivalent to $${z^n}$$ . When the order of $${\phi}$$ is 4, we determine the reducing subspaces for $${M_\phi}$$ , and we see that in this case $${M_\phi}$$ can be reducible on D when $${\phi}$$ is not equivalent to $${z^4}$$ . The same phenomenon happens when the order n of $${\phi}$$ is not a prime number. Furthermore, we show that $${M_\phi}$$ is unitarily equivalent to $${M_{z^n} (n > 1)}$$ on D if and only if $${\phi = az^n}$$ for some unimodular constant a.

Generalized bundle shift with application to multiplication operator on the Bergman space

Following upon results of Putinar, Sun, Wang, Zheng and the first author, we provide models for the restrictions of the multiplication by a finite Balschke product on the Bergman space in the unit disc to its reducing subspaces. The models involve a generalization of the notion of bundle shift on the Hardy space introduced by Abrahamse and the first author to the Bergman space. We develop generalized bundle shifts on more general domains. While the characterization of the bundle shift is rather explicit, we have not been able to obtain all the earlier results appeared, in particular, the facts that the number of the minimal reducing subspaces equals the number of connected components of the Riemann surface $B(z)=B(w)$ and the algebra of commutant of $T_{B}$ is commutative, are not proved. Moreover, the role of the Riemann surface is not made clear also.

Reducing subspaces of tensor products of weighted shifts

A unilateral weighted shift A is said to be simple if its weight sequence {αn} satisfies ∇3(αn2) ≠ 0 for all n ≥ 2. We prove that if A and B are two simple unilateral weighted shifts, then A⊗I + I ⊗B is reducible if and only if A and B are unitarily equivalent. We also study the reducing subspaces of Ak ⊗ I + I ⊗ Bl and give some examples. As an application, we study the reducing subspaces of multiplication operators \(M_{z^k + \alpha w^l }\) on function spaces.

Reducing Subspaces of Some Multiplication Operators on the Bergman Space over Polydisk

We consider the reducing subspaces ofMzNonAα2(Dk), wherek≥3,zN=z1N1⋯zkNk, andNi≠Njfori≠j. We prove that each reducing subspace ofMzNis a direct sum of some minimal reducing subspaces. We also characterize the minimal reducing subspaces in the cases thatα=0andα∈(-1,+∞)∖Q, respectively. Finally, we give a complete description of minimal reducing subspaces ofMzNonAα2(D3)withα>-1.

Weak (quasi-)affine bi-frames for reducing subspaces of L 2(ℝ d )

Since a frame for a Hilbert space must be a Bessel sequence, many results on (quasi-)affine bi-frame are established under the premise that the corresponding (quasi-)affine systems are Bessel sequences. However, it is very technical to construct a (quasi-)affine Bessel sequence. Motivated by this observation, in this paper we introduce the notion of weak (quasi-)affine bi-frame (W(Q)ABF) in a general reducing subspace for which the Bessel sequence hypothesis is not needed. We obtain a characterization of WABF, and prove the equivalence between WABF and WQABF under a mild condition. This characterization is used to recover some related known results in the literature.

Geometric constructions of thin Blaschke products and reducing subspace problem

In this paper, we mainly study geometric constructions of thin Blaschke products B and reducing subspace problem of multiplication operators induced by such symbols B on the Bergman space. Considering such multiplication operators M B , we present a representation of those operators commuting with both M B and M B * . It is shown that for ‘most’ thin Blaschke products B, M B is irreducible, that is, M B has no nontrivial reducing subspace; and such a thin Blaschke product B is constructed. As an application of the methods, it is proved that for ‘most’ finite Blaschke products ϕ, M ϕ has exactly two minimal reducing subspaces. Furthermore, under a mild condition, we get a geometric characterization for when M B defined by a thin Blaschke product B has a nontrivial reducing subspace.

REDUCING SUBSPACES FOR TOEPLITZ OPERATORS ON THE POLYDISK

In this note, we completely characterize the reducing sub- spaces of T zN zM on A 2 (D 2) where � > −1 and N,M are positive in- tegers with N 6 M, and show that the minimal reducing subspaces of T zN zM on the unweighted Bergman space and on the weighted Bergman space are different.

Inner envelopes: efficient estimation in multivariate linear regression

In this article we propose a new model, called the inner envelope model, which leads to efficient estimation in the context of multivariate normal linear regression. The asymptotic distribution and the consistency of its maximum likelihood estimators are established. Theoretical results, simulation studies and examples all show that the efficiency gains can be substantial relative to standard methods and to the maximum likelihood estimators from the envelope model introduced recently by Cook et al. (2010). Compared to the envelope model, the inner envelope model is based on a different construction and it can produce substantial efficiency gains in situations where the envelope model offers no gains. In effect, inner envelopes open a new frontier to the way in which reducing subspaces can be used to improve efficiency in multivariate problems. Copyright 2012, Oxford University Press.

Reducing subspaces for analytic multipliers of the Bergman space

In Douglas et al. (2011) [4] some incisive results are obtained on the structure of the reducing subspaces for the multiplication operator Mϕ by a finite Blaschke product ϕ on the Bergman space on the unit disk. In particular, the linear dimension of the commutant, Aϕ={Mϕ,Mϕ⁎}′, is shown to equal the number of connected components of the Riemann surface, ϕ−1∘ϕ. Using techniques from Douglas et al. (2011) [4] and a uniformization result that expresses ϕ as a holomorphic covering map in a neighborhood of the boundary of the disk, we prove that Aϕ is commutative, and moreover, that the minimal reducing subspaces are pairwise orthogonal. Finally, an analytic/arithmetic description of the minimal reducing subspaces is also provided, along with the taxonomy of the possible structures of the reducing subspaces in case ϕ has eight zeros. These results have implications in both operator theory and the geometry of finite Blaschke products.

Invariant subspaces and reducing subspaces of weighted Bergman space over polydisc

In this paper, we study the invariant subspaces and reducing subspace of the weighted Bergman space over polydisc. The minimal reducing subspaces of a class of analytic Toeplitz operators are completely described, and Beurling-type theorem of some invariant subspaces of Toeplitz operators T zi (1≤ i ≤ n )on the weighted Bergman space over polydisc is also obtained.

Multiplication operators on the Bergman space via analytic continuation

In this paper, using the group-like property of local inverses of a finite Blaschke product ϕ, we will show that the largest C⁎-algebra in the commutant of the multiplication operator Mϕ by ϕ on the Bergman space is finite dimensional, and its dimension equals the number of connected components of the Riemann surface of ϕ−1∘ϕ over the unit disk. If the order of the Blaschke product ϕ is less than or equal to eight, then every C⁎-algebra contained in the commutant of Mϕ is abelian and hence the number of minimal reducing subspaces of Mϕ equals the number of connected components of the Riemann surface of ϕ−1∘ϕ over the unit disk.

Invariant subspaces and reducing subspaces of weighted Bergman space over bidisk

In this paper, we study the invariant subspace and reducing subspace of the weighted Bergman space over bidisk. The minimal reducing subspace of Toeplitz operator T z N = T z 1 N z 2 N is completely described, and Beurling-type theorem of some invariant subspace of the weighted Bergman space over bidisk is also obtained.

Classification of Reducing Subspaces of a Class of Multiplication Operators on the Bergman Space via the Hardy Space of the Bidisk

AbstractIn this paper we obtain a complete description of nontrivial minimal reducing subspaces of the multiplication operator by a Blaschke product with four zeros on the Bergman space of the unit disk via the Hardy space of the bidisk.

Invariant subspaces of parabolic self-maps in the Hardy space

We provide a precise description of the lattice of invariant subspaces of composition operators acting on the classical Hardy space, whose inducing symbol is a parabolic non-automorphism. This is achieved with an explicit isomorphism between the Hardy space and the Sobolev Banach algebra $W^{1,2}[0,\infty)$ that induces a bijection between the lattice of the composition operator and the closed ideals of $W^{1,2}[0,\infty)$. In particular, each invariant subspace of parabolic non-automorphism composition operator always consists of the closed span of a set of eigenfunctions. As a consequence, such composition operators have no non-trivial reducing subspaces. For the sake of completeness, we also include a characterization of the closed ideals of the Banach algebra $W^{1,2}[0,\infty)$. Although such a characterization is known, the proof we provide here is somehow different.

Orthogonal decompositions for wavelets

Decompositions of Hilbert spaces in terms of reducing subspaces for wavelets operators, as well decompositions of these operators themselves, are investigated. In particular, it is shown on which reducing subspaces these operators act as bilateral shifts of multiplicity 1. We also exhibit the unitary transformation that performs the unitary equivalence between restrictions of them to appropriate reducing subspaces.

Shift reducing subspaces and irreducible-invariant subspaces generated by wandering vectors and applications

We introduce the notions of elementary reducing subspaces and elementary irreducible-invariant subspaces-- generated from wandering vectors--of a shift operator of countably infinite multiplicity, defined on a separable Hilbert space H. Necessary and sufficient conditions for a set of shift wandering vectors to span a wandering subspace are obtained. These lead to characterizations of shift reducing subspaces and shift irreducible-invariant subspaces, as well as a new decomposition of H into orthogonal sum of elementary reducing subspaces. Applications of elementary reducing subspaces to wavelet expansion, and of elementary irreducible-invariant subspaces to wavelet multiresolution analysis (MRA) will be discussed.

Frame wavelets in subspaces of $L^2(\mathbb R^d)$

Let $A$ be a $d\times d$ real expansive matrix. We characterize the reducing subspaces of $L^2(\mathbb {R}^d)$ for $A$-dilation and the regular translation operators acting on $L^2 (\mathbb {R}^d).$ We also characterize the Lebesgue measurable subsets $E$ of $\mathbb {R}^d$ such that the function defined by inverse Fourier transform of $[1/(2\pi )^{d/2}]\chi _{E}$ generates through the same $A$-dilation and the regular translation operators a normalized tight frame for a given reducing subspace. We prove that in each reducing subspace, the set of all such functions is nonempty and is also path connected in the regular $L^2(\mathbb {R}^d)$-norm.

Reducing subspaces of compressed analytic Toeplitz operators on the Hardy space

In this paper we discuss necessary conditions and sufficient conditions for the compression of an analytic Toeplitz operator onto a shift coinvariant subspace to have nontrivial reducing subspaces. We give necessary and sufficient conditions for the kernel of a Toeplitz operator whose symbol is the quotient of two inner functions to be nontrivial and obtain examples of reducing subspaces from these kernels. Motivated by this result we give necessary conditions and sufficient conditions for the kernel of a Toeplitz operator whose symbol is the quotient of two inner functions to be nontrivial in terms of the supports of the two inner functions. By studying the commutant of a compression, we are able to give a necessary condition for the existence of reducing subspaces on certain shift coinvariant subspaces.

An Algorithm for Computing Reducing Subspaces by Block Diagonalization

This paper describes an algorithm for reducing a real matrix A to block diagonal form by a real similarity transformation. The columns of the transformation corresponding to a block span a reducing subspace of A, and the block is the representation of A in that subspace with respect to the basis. The algorithm attempts to control the condition of the transformation matrices, so that the reducing subspaces are well conditioned and the basis vectors are numerically independent.

The set of irreducible operators is dense

Let 3C be a separable (finite or infinite-dimensional) complex Hilbert space. P. R. Halmos [1] has shown that the set of irreducible operators on 3C, (i.e., operators with no nontrivial reducing subspaces), is uniformly dense in the space of bounded operators on SC. In this note we give a very simple proof of Halmos's theorem. Let A be any bounded operator on ae and let e > 0. By the spectral theorem there exists a Hermitian operator D whose matrix is diagonal with respect to an o.n. basis { en } such that