In this paper we study the kernels of Toeplitz operators on both the scalar and the vector-valued Hardy space for $ 1 < p < \infty $. We show existence of a minimal kernel of any element of the vector-valued Hardy space and we determine a symbol for the corresponding Toeplitz operator. In the scalar case we give an explicit description of a maximal function for a given Toeplitz kernel. In the vectorial case we show not all Toeplitz kernels have a maximal function and in the case of $p=2$ we find the exact conditions for when a Toeplitz kernel has a maximal function. For both the scalar and vector-valued Hardy space we study the minimal Toeplitz kernel containing multiple elements of the Hardy space, we also find an equivalent condition for a function in the Smirnov class to be cyclic for the backwards shift.
Recently, Liang and Partington \cite{YP} show that kernels of finite-rank perturbations of Toeplitz operators are nearly invariant with finite defect under the backward shift operator acting on the scalar-valued Hardy space. In this article we provide a vectorial generalization of a result of Liang and Partington. As an immediate application we identify the kernel of perturbed Toeplitz operator in terms of backward shift-invariant subspaces in various important cases by applying the recent theorem (\cite{CDP, OR}) in connection with nearly invariant subspaces of finite defect for the backward …
It is well known that the kernel of a Toeplitz operator is nearly invariant under the backward shift $S^*$. This paper shows that kernels of finite-rank perturbations of Toeplitz operators are nearly $S^*$-invariant with finite defect. This enables us to apply a recent theorem by Chalendar--Gallardo--Partington to represent the kernel in terms of backward shift-invariant subspaces, which we identify in several important cases.
Compressions of Toeplitz operators to coinvariant subspaces of $H^2$ are called truncated Toeplitz operators. We study two questions related to these operators. The first, raised by Sarason, is whether boundedness of the operator implies the existence of a bounded symbol; the second is the reproducing kernel thesis. We show that in general the answer to the first question is negative, and we exhibit some classes of spaces for which the answers to both questions are positive.
Toeplitz plus Hankel operators $T(a)+H(b)$, $a,b\in L^\infty$ acting on the classical Hardy spaces $H^p, 1<p<\infty$, are studied. If the generating functions $a$ and $b$ satisfy the so-called matching condition $a(t) a(1/t)=b(t) b(1/t)$, an effective description of the structure of the kernel and cokernel of the corresponding operator is given. The results depend on the behaviour of two auxiliary scalar Toeplitz operators, and if the generating functions $a$ and $b$ are piecewise continuous, more detailed results are obtained.
Asymmetric dual truncated Toeplitz operators acting between the orthogonal complements of two (eventually different) model spaces are introduced and studied. They are shown to be equivalent after extension to paired operators on $L^2(\mathbb T) \oplus L^2(\mathbb T)$ and, if their symbols are invertible in $L^\infty(\mathbb T)$, to asymmetric truncated Toeplitz operators with the inverse symbol. Relations with Carleson's corona theorem are also established. These results are used to study the Fredholmness, the invertibility and the spectra of various classes of dual truncated Toeplitz operator…
The paper is concerned with a sequence of constants which appear in several problems. These problems include the minimal eigenvalue of certain positive definite Toeplitz matrices, the minimal eigenvalue of some higher-order ordinary differential operators, the norm of the Green kernels of these operators, the best constant in a Wirtinger-Sobolev inequality, and the conditioning of a special least squares problem. The main result of the paper gives the asymptotics of this sequence.
This paper studies matrix-valued truncated Toeplitz operators, which are a vectorial generalisation of truncated Toeplitz operators. It is demonstrated that, although there exist matrix-valued truncated Toeplitz operators without a matrix symbol in $L^p$ for any $p \in (2, \infty ]$, there is a wide class of matrix-valued truncated Toeplitz operators which possess a matrix symbol in $L^p$ for some $p \in (2, \infty ]$. In the case when the matrix-valued truncated Toeplitz operator has a symbol in $L^p$ for some $p \in (2, \infty ]$, an approach is developed which bypasses some of the technical…
In this paper we consider compressions of $k^{th}$--order slant Toeplitz operators to the backward shift invariant subspaces of the classical Hardy space $H^2$. In particular, we characterize these operators using compressed shifts and finite rank operators of special kind.
Generalized Toeplitz plus Hankel operators $T(a)+H_α(b)$ generated by functions $a,b$ and a linear fractional Carleman shift $α$ changing the orientation of the unit circle $\mathbb{T}$ are considered on the Hardy spaces $H^p(\mathbb{T})$, $1<p<\infty$. If the functions $a,b\in L^\infty(\mathbb{T})$ and satisfy the condition $$ a(t) a(α(t))=b(t) b(α(t)),\quad t\in \mathbb{T}, $$ the defect numbers of the operators $T(a)+H_α(b)$ are established and an explicit description of the structure of the kernels and cokernels of the operators mentioned is given.
We review some classical and more recent results concerning kernels of Toeplitz operators and their relations with model spaces, which are themselves Toeplitz kernels of a special kind. We highlight the fundamental role played by the existence of maximal vectors for every nontrivial Toeplitz kernel.
Truncated Toeplitz operators are compressions of multiplication operators on $L^2$ to model spaces (that is, subspaces of $H^2$ which are invariant with respect to the backward shift). For this class of operators we prove certain Szegö type theorems concerning the asymptotics of their compressions to an increasing chain of finite dimensional model spaces.
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 factorizations. 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.
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-vani…
We characterize operator-theoretic properties (boundedness, compactness, and Schatten class membership) of Toeplitz operators with positive measure symbols on Bergman spaces of holomorphic hermitian line bundles over Kähler Cartan-Hadamard manifolds in terms of geometric or operator-theoretic properties of measures.
This paper considers model spaces in an $H_p$ setting. The existence of unbounded functions and the characterisation of maximal functions in a model space are studied, and decomposition results for Toeplitz kernels, in terms of model spaces, are established.
In this note, we describe the backward shift invariant subspaces for a large class of reproducing kernel Hilbert spaces. This class includes in particular de Branges-Rovnyak spaces (the non-extreme case) and the range space of co-analytic Toeplitz operators.
It is shown that the kernel of a Toeplitz operator with $2\times 2$ symbol $G$ can be described exactly in terms of any given function in a very wide class, its image under multiplication by $G$, and their left inverses, if the latter exist. As a consequence, under many circumstances the kernel of a block Toeplitz operator may be described as the product of a space of scalar complex-valued functions by a fixed column vector of functions. Such kernels are said to be of scalar type, and in this paper they are studied and described explicitly in many concrete situations. Applications are given to…
We apply the theory of de Branges-Rovnyak spaces to describe kernels of some Toeplitz operators on the classical Hardy space $H^2$. In particular, we discuss the kernels of the operators $T_{\bar f/ f}$ and $T_{\bar I\bar f/ f}$, where $f$ is an outer function in $H^2$ and $I$ is inner such that $I(0)=0$. We also obtain results on de Branges-Rovnyak spaces generated by nonextreme functions.
The classical theory of Toeplitz operators in spaces of analytic functions deals usually with symbols that are bounded measurable functions on the domain in question. A further extension of the theory was made for symbols being unbounded functions, measures, and compactly supported distributions, all of them subject to some restrictions. In the context of a reproducing kernel Hilbert space we propose a certain framework for a `maximally possible' extension of the notion of Toeplitz operators for a `maximally wide' class of `highly singular' symbols. Using the language of sesquilinear forms we …
For a smoothly bounded strictly pseudoconvex domain, we describe the boundary singularity of weighted Bergman kernels with respect to weights behaving like a power (possibly fractional) of a defining function, and, more generally, of the reproducing kernels of Sobolev spaces of holomorphic functions of any real order. This generalizes the classical result of Fefferman for the unweighted Bergman kernel. Finally, we also exhibit a holomorphic continuation of the kernels with respect to the Sobolev parameter to the entire complex plane. Our main tool are the generalized Toeplitz operators of Bout…
Let $m \geq 1$ be an integer and let $H_m(\mathbb B)$ be the analytic functional Hilbert space on the unit ball $\mathbb B \subset \mathbb C^n$ given by the reproducing kernel $K_m(z,w) = (1 - \langle z,w \rangle)^{-m}$. We prove that Toeplitz operators with pluriharmonic symbol on $H_m(\mathbb B)$ can be characterized by an algebraic identity which extends the classical Brown-Halmos characterization of Toeplitz operators on the Hardy space of the unit disc as well as corresponding results of Louhichi and Olofsson for Toeplitz operators with harmonic symbol on weighted Bergman spaces of the di…
Let $L^2$ be the Lebesgue space of square-integrable functions on the unit circle. We show that the injectivity problem for Toeplitz operators is linked to the existence of geodesics in the Grassmann manifold of $L^2$. We also investigate this connection in the context of restricted Grassmann manifolds associated to $p$-Schatten ideals and essentially commuting projections.
We define and study Toeplitz operators in the space of Herglotz solutions of the Helmholtz equation in $R^d$. As the most traditional definition of Toeplitz operators via Bergman-type projection is not available here, we use an approach based upon the reproducing kernel nature of the Herglotz space and sesquilinear forms, which results in a meaningful theory. For two important patterns of sesquilinear forms we discuss a number of properties, including the uniqueness of determining the symbols from the operator, the finite rank property, the conditions for boundedness and compactness, spectral …
In this paper we first study the structure of the scalar and vector-valued nearly invariant subspaces with a finite defect. We then subsequently produce some fruitful applications of our new results. We produce a decomposition theorem for the vector-valued nearly invariant subspaces with a finite defect. More specifically, we show every vector-valued nearly invariant subspace with a finite defect can be written as the isometric image of a backwards shift invariant subspace. We also show that there is a link between the vector-valued nearly invariant subspaces and the scalar-valued nearly invar…
Let $S$ be the shift operator on the Hardy space $H^2$ and let $S^*$ be its adjoint. A closed subspace $\FF$ of $H^2$ is said to be nearly $S^*$-invariant if every element $f\in\FF$ with $f(0)=0$ satisfies $S^*f\in\FF$. In particular, the kernels of Toeplitz operators are nearly $S^*$-invariant subspaces. Hitt gave the description of these subspaces. They are of the form $\FF=g (H^2\ominus u H^2)$ 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 func…
Let $Γ$ be a rectifiable Jordan curve, let $X$ and $Y$ be two reflexive Banach function spaces over $Γ$ such that the Cauchy singular integral operator $S$ is bounded on each of them, and let $M(X,Y)$ denote the space of pointwise multipliers from $X$ to $Y$. Consider the Riesz projection $P=(I+S)/2$, the corresponding Hardy type subspaces $PX$ and $PY$, and the Toeplitz operator $T(a):PX\to PY$ defined by $T(a)f=P(af)$ for a symbol $a\in M(X,Y)$. We show that if $X\hookrightarrow Y$ and $a\in M(X,Y)\setminus\{0\}$, then $T(a)\in\mathcal{L}(PX,PY)$ has a trivial kernel in $PX$ or a dense image…
For bounded Lebesgue measurable functions $f,g,ϕ$ and $ψ$ on the unit circle, $P_{+}fP_{+}+P_{-}gP_{+} +P_{+}ϕP_{-}+P_{-}ψP_{-}$ is called a generalized singular integral operator (GSIO) on $L^{2}(\mathbb{T})$, where $P_{+}$ is the Riesz projection, $P_{-}=I-P_{+}.$ In this paper, we relate GSIOs to a number of operators, including Cauchy singular integral operator, (dual) truncated Toeplitz operator, Foguel-Hankel operator, multiplication operator, Toeplitz plus Hankel operator etc. We establish the short exact sequences associated of the $C^{*}-$algebras generated by GSIOs with bounded or qu…
We study idempotent, model, and Toeplitz operators that attain the norm. Notably, we prove that if $\mathcal{Q}$ is a backward shift invariant subspace of the Hardy space $H^2(\mathbb{D})$, then the model operator $S_{\mathcal{Q}}$ attains its norm. Here $S_{\mathcal{Q}} = P_{\mathcal{Q}}M_z|_{\mathcal{Q}}$, the compression of the shift $M_z$ on the Hardy space $H^2(\mathbb{D})$ to $\mathcal{Q}$.
It is well known that a Toeplitz operator is invertible if and only if its symbols admits a canonical Wiener-Hopf factorization, where the factors satisfy certain conditions. A similar result holds also for singular integral operators. More general, the dimension of the kernel and cokernel of Toeplitz or singular integral operators which are Fredholm can be expressed in terms of the partial indices of an associated Wiener-Hopf factorization problem. In this paper we establish corresponding results for Toeplitz + Hankel operators and singular integral operators with flip under the assumption th…
We introduce new tools for analytic microlocal analysis on Kähler manifolds. As an application, we prove that the space of Berezin-Toeplitz operators with analytic contravariant symbol is an algebra. We also give a short proof of the Bergman kernel asymptotics up to an exponentially small error.
In this paper we study mapping properties of Toeplitz-like operators on weighted Bergman spaces of bounded strongly pseudconvex domains in $\mathbb{C}^n$. In particular we prove that a Toeplitz operator built using as kernel a weighted Bergman kernel of weight $β$ and integrating against a measure $μ$ maps continuously (when $β$ is large enough) a weighted Bergman space $A^{p_1}_{α_1}(D)$ into a weighted Bergman space $A^{p_2}_{α_2}(D)$ if and only if $μ$ is a $(λ,γ)$-skew Carleson measure, where $λ=1+\frac{1}{p_1}-\frac{1}{p_2}$ and $γ=\frac{1}λ\left(β+\frac{α_1}{p_1}-\frac{α_2}{p_2}\right)$.…
We show that the same similarity characterization obtained for Cowen-Douglas operators to the backward shift operators on reproducing kernel Hilbert spaces with analytic kernels can be used to describe similarity in the Dirichlet space setting. As in previous proofs, a model theorem that allows one to get the eigenvector bundle structure of the operator plays a crucial role.
Unbounded (and bounded) Toeplitz operators (TO) with rational symbols are analysed in detail showing that they are densely defined closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains, ranges, spectral and Fredholm points are determined. In particular, in the symmetric case, i.e., for a real rational symbol the deficiency spaces and indices are explicitly available. The concluding section gives a brief overview on the research on unbounded TO in order to locate the present contribution. Regarding properties of unbounded TO in general, it fu…
The symmetrized bidisc has been a rich field of holomorphic function theory and operator theory. A certain well-known reproducing kernel Hilbert space of holomorphic functions on the symmetrized bidisc resembles the Hardy space of the unit disc in several aspects. This space is known as the Hardy space of the symmetrized bidisc. We introduce the study of those operators on the Hardy space of the symmetrized bidisc that are analogous to Toeplitz operators on the Hardy space of the unit disc. More explicitly, we first study multiplication operators on a bigger space (an $L^2$-space) and then stu…
The Invariant Subspace Problem (ISP) for Hilbert spaces asks if every bounded linear operator has a non-trivial closed invariant subspace. Due to the existence of universal operators (in the sense of Rota), the ISP may be solved by describing the invariant subspaces of these operators alone. We characterize all anaytic Toeplitz operators $T_ϕ$ on the Hardy space $H^2(\mathbb{D}^n)$ over the polydisk $\mathbb{D}^n$ for $n>1$ whose adjoints satisfy the Caradus criterion for universality, that is, when $T_ϕ^*$ is surjective and has infinite dimensional kernel. In particular if $ϕ$ in a non-consta…
