Kernels Of Toeplitz Operators Research Articles

Conjugations Preserving Toeplitz Kernels

We study conjugations in L^2(mathbb {T}) and their relation with kernels of Toeplitz operators on H^2(mathbb {T}) space. Such kernels are a generalization of model spaces. We investigate properties of an inequality relation between two unimodular functions defined on the unit circle. This allows us to significantly strengthen previous theorems characterizing all M_z-commuting and M_z-conjugations leaving invariant model spaces to the Toeplitz kernels setting.

Finite zero-based invariant subspaces of the shift operator on reproducing kernel spaces

We characterize wandering subspace property of the shift operator on zero-based invariant subspaces of reproducing kernel Hilbert spaces by using the kernels of Toeplitz operators. As applications, we prove that the wandering subspace property of the shift operator holds on the weighted Bergman space () for zero-based invariant subspaces when the zeros lie in a small disk centred at the origin.

The Riemann Zeta Function and Kernels of Toeplitz Operators

The Riemann Zeta Function and Kernels of Toeplitz Operators

Kernels of Unbounded Toeplitz Operators and Factorization of Symbols

We consider kernels of unbounded Toeplitz operators in H^p({mathbb {C}}^{+}) in terms of a factorization of their symbols. We study the existence of a minimal Toeplitz kernel containing a given function in H^p({mathbb {C}}^{+}), we describe the kernels of Toeplitz operators whose symbol possesses a certain factorization involving two different Hardy spaces and we establish relations between the kernels of two operators whose symbols differ by a factor which corresponds, in the unit circle, to a non-integer power of z. We apply the results to describe the kernels of Toeplitz operators with non-vanishing piecewise continuous symbols.

Kernels of Toeplitz Operators and Rational Interpolation

The kernel of a Toeplitz operator on the Hardy class H2 in the unit disk is a nearly invariantsubspace of the backward shift operator, and, by D. Hitt’s result, it has the form g · Kω where ω is an inner function, Kω = H2 ⊝ ωH2, and g is an isometric multiplier on Kω. We describe the functions ω and g for the kernel of the Toeplitz operator with symbol .$$ \overline{\theta}\varDelta $$ where θ is an inner function and Δ is a finite Blaschke product.

Multipliers and equivalences between Toeplitz kernels

Multipliers between kernels of Toeplitz operators are characterised in terms of test functions (so-called maximal vectors for the kernels); these maximal vectors may easily be parametrised in terms of inner and outer factorisations. Immediate applications to model spaces are derived. The case of surjective multipliers is also analysed. These ideas are applied to describing equivalences between two Toeplitz kernels.

Kernels of Toeplitz operators on the Hardy space over the bidisk

We study the kernels of Toeplitz operators on the Hardy space on the bidisk. We first give a sufficient condition for a general symbol to be antiholomorphic under the assumption that the kernel of the corresponding Toeplitz operator contains a backward shift invariant subspace. As an application, we construct an invariant subspace whose its orthogonal complement can not be the kernel of any Toeplitz operator. Also, we give a characterization on a Toeplitz operator for which the orthogonal complement of its kernel is generated by certain inner functions. Finally, we describe all backward shift invariant subspaces which are in the kernels of Toeplitz operators with homogeneous type symbols. As an application, we show that there is a Toeplitz operator for which its kernel has dimension of any given integer and the orthogonal complement of the kernel is generated by two functions. Our result shows that there are higher dimensional phenomena.

Some remarks on analytic continuation in weighted backward shift invariant subspaces

By a famous result of Douglas, Shapiro, and Shields, functions in backward shift invariant subspaces in Hardy spaces are characterized by the fact that they admit a pseudocontinuation outside the closed unit disk. More can be said when the spectrum of the associated inner function has holes on \({{\mathbb T}}\). Then the functions of the invariant subspaces even extend analytically through these holes. Here we will be interested in weighted backward shift invariant subspaces which appear naturally in the context of kernels of Toeplitz operators. Note that such kernels are special cases of so-called nearly invariant subspaces. In our setting a result by Aleksandrov allows to deduce analytic continuation properties which we will then apply to consider embeddings of weighted invariant subspaces into their unweighted companions. We hope that this connection might shed some new light on known results. We will also establish a link between the spectrum of the inner function and the approximate point spectrum of the backward shift in the weighted situation in the spirit of results by Aleman, Richter, and Ross.

Abstract Interpolation in Vector-Valued de Branges–Rovnyak Spaces

Following ideas from the Abstract Interpolation Problem of Katsnelson et al. (Operators in spaces of functions and problems in function theory, vol 146, pp 83-69, Naukova Dumka, Keiv, 1987) for Schur class functions, we study a general metric constrained interpolation problem for functions from a vector-valued de Branges-Rovnyak space $\mathcal{H}(K_S)$ associated with an operator-valued Schur class function $S$. A description of all solutions is obtained in terms of functions from an associated de Branges-Rovnyak space satisfying only a bound on the de Branges-Rovnyak-space norm. Attention is also paid to the case that the map which provides this description is injective. The interpolation problem studied here contains as particular cases (1) the vector-valued version of the interpolation problem with operator argument considered recently in Ball et al. (Proc Am Math Soc 139(2), 609-618, 2011) (for the nondegenerate and scalar-valued case) and (2) a boundary interpolation problem in $\mathcal{H}(K_S)$. In addition, we discuss connections with results on kernels of Toeplitz operators and nearly invariant subspaces of the backward shift operator.

Kernel of Vector-Valued Toeplitz Operators

Let S be the shift operator on the Hardy space H2 and let S* be its adjoint. A closed subspace \({\mathcal F}\) of H2 is said to be nearly S*-invariant if every element \({f\in\mathcal F}\) with f(0) = 0 satisfies \({S^*f\in\mathcal F}\). In particular, the kernels of Toeplitz operators are nearly S*-invariant subspaces. Hitt gave the description of these subspaces. They are of the form \({\mathcal F=g (H^2\ominus u H^2)}\) with \({g\in H^2}\) and u inner, u(0) = 0. A very particular fact is that the operator of multiplication by g acts as an isometry on \({H^2\ominus uH^2}\). Sarason obtained a characterization of the functions g which act isometrically on \({H^2\ominus uH^2}\). Hayashi obtained the link between the symbol \({\varphi}\) of a Toeplitz operator and the functions g and u to ensure that a given subspace \({\mathcal F=gK_u}\) is the kernel of \({T_\varphi}\). Chalendar, Chevrot and Partington studied the nearly S*-invariant subspaces for vector-valued functions. In this paper, we investigate the generalization of Sarason’s and Hayashi’s results in the vector-valued context.

Kernels of Toeplitz Operators via Bourgain's Factorization Theorem

Using Bourgain's factorization theorem, we characterize the subspaces of Hp, 1⩽p⩽∞, that coincide with the kernels of Toeplitz operators. This is related to (but entirely independent of) earlier work of E. Hayashi. One consequence of our characterization is that, for some inner functions θ, the class of uniqueness sets for Kpθ depends on p; here Kpθ=defHp∩θzHp is the star-invariant subspace in Hp generated by θ. Moreover, a construction is provided of Blaschke products B and b for which the dimension function p↦dim(KpB∩bHp) has prescribed jumps at prescribed points.

Interpolating Functions of Minimal Norm, Star-Invariant Subspaces, and Kernels of Toeplitz Operators

Interpolating Functions of Minimal Norm, Star-Invariant Subspaces, and Kernels of Toeplitz Operators

Interpolating functions of minimal norm, star-invariant subspaces and kernels of Toeplitz operators

It is proved that for each inner function θ \theta there exists an interpolating sequence { z n } \left \{ {{z_n}} \right \} in the disk such that sup n | θ ( z n ) | > 1 {\sup _n}|\theta ({z_n})| > 1 , but every function g g in H ∞ {H^\infty } with g ( z n ) = θ ( z n ) ( n = 1 , 2 , … ) g({z_n}) = \theta ({z_n})(n = 1,2, \ldots ) satisfies | | g | | ∞ ≥ 1 ||g|{|_\infty } \geq 1 . Some results are obtained concerning interpolation in the star-invariant subspace H 2 ⊖ θ H 2 {H^2} \ominus \theta {H^2} . This paper also contains a "geometric" result connected with kernels of Toeplitz operators.

Classification of nearly invariant subspaces of the backward shift

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.

Kernels of Toeplitz operators

Kernels of Toeplitz operators