Kernel of Vector-Valued Toeplitz Operators

Let S ∗ {S^*} denote the backward shift operator on the Hardy space H 2 {H^2} of the unit disk. A subspace M M of H 2 {H^2} is called nearly invariant if S ∗ h {S^*}h is in M M whenever h h belongs to M M and h ( 0 ) = 0 h(0) = 0 . In particular, the kernel of every Toeplitz operator is nearly invariant. A function theoretic characterization is given of those nearly invariant subspaces which are the kernels of Toeplitz operators, and it is shown that they can be put into one-to-one correspondence with the Cartesian product of the set of exposed points of the unit ball of H 1 {H^1} with the set of inner functions.

For a compact subset K in the complex plane, let A(K) denote the algebra of all functions continuous on K and analytic on K° and let R(K) denote the uniform closure of the rational functions with poles off K. Let G is a bounded open subset whose complement in the plane has a finite number of components. Suppose that $$A(\overline{G} ) = R(\overline{G} )$$ and every function in H∞(G) is the pointwise limit of a bounded sequence of functions in $$R(\overline{G} )$$ . The purpose of this paper is to characterize all subnormal operators similar to M z , the operator of multiplication by the independent variable z on the Hardy space H2(G). We also characterize all bounded linear operators that are unitarily equivalent to M z in the case when each of the components of G is simply connected. In particular, our similarity result extends a well-known result of W. Clary on the unit disk to multiply connected domains.

Multipliers and equivalences between Toeplitz kernels

We present a unified approach to the problem of characterizing the Fredholm multiplication and composition operators on the Hardy spaceH 2.

Toeplitz and asymptotic Toeplitz operators on [formula omitted

Let M be a forward-shift-invariant subspace and N a backward-shift-invariant subspace in the Hardy space H2 on the bidisc. We assume that $H^2=N \oplus M$ . Using the wandering subspace of M and N, we study the relations between M and N. Moreover we study M and N using several natural operators defined by shift operators on H2.

Extremal functions as divisors for kernels of Toeplitz operators

There is axiomatically described the class of spaces Υ (resp. Χ) of functions, analytic in the unit disk, for which the invariant subspaces of the shift operator f (z) → z f (z) (resp. the inverse shift f(z)→z−1(f(z)−f (0))) are constructed just like the Hardy space H2. It is proved that as Χ one can take, for example, the space H1, the disk-algebra CA, the space UA of all uniformly convergent power series; and as Υ the space of integrals of Cauchy type L1/H 1− , the space VMOA. There is also obtained an analog for the space UA of W. Rudin's theorem on z-invariant subspaces of the space CA.

Given a unilateral forward shift S acting on a complex, separable, innite dimensional Hilbert space H, an asymptotically S-Toeplitz operator is a bounded linear operator T on H satisfying that {S* n TS n } is convergent with respect to one of the topologies commonly used in the algebra of bounded linear operators on H. In this paper, we study the asymptotic T u -Toeplitzness of weighted composition operators on the Hardy space H2, where u is a nonconstant inner function.

Shift-type invariant subspaces of contractions

Konstantin M. Dyakonov, Interpolating Functions of Minimal Norm, Star-Invariant Subspaces, and Kernels of Toeplitz Operators, Proceedings of the American Mathematical Society, Vol. 116, No. 4 (Dec., 1992), pp. 1007-1013

An estimate of the spectral radius of functional operators generated by operators of multiplication and shift operators in the space of continuous vector functions on the interval is given. It is assumed that shifts have fixed points only at both ends of the interval and belong to one type, i.e., they are either left or right shifts.

Let M_z and B denote, respectively, the multiplication operator and the backward shift operator on a weighted Hardy space. We present sufficient conditions so that M_{z^n} is similar to \bigoplus_1^nM_z, and B^n is similar to \bigoplus_1^nB. The first part generalizes a result obtained by Yucheng Li.

For a large class of Hilbert spaces of analytic functions in the unit disc lying between the Hardy and the Dirichlet space we prove that each element of the space is the quotient of two bounded functions in the same space. It follows that the multiplication operator on these spaces is cellular indecomposable and that each invariant subspace contains nontrivial bounded functions.

In this paper we study the Hilbert space of analytic functions with finite Dirichlet integral in a connected open set C2 in the complex plane. We show that every such function can be represented as a quotient of two bounded analytic functions, each of which has a finite Dirichlet integral. This has several consequences for the structure of invariant subspaces of the algebra of multiplication operators on the Dirichlet space, in case f2 is simply connected. Namely, we show that every nontrivial invariant subspace contains a nontrivial bounded function, that each two nontrivial invariant subspaces have a nontrivial intersection (that is, the algebra is cellular indecomposable), and that each nontrivial invariant subspace has the codimension one property with respect to certain multiplication operators.