Related papers: Abrahamse's Theorem for matrix-valued symbols and …

Suppose that $\phi$ and $\psi$ are smooth complex-valued functions on the circle that are invertible, have winding number zero with respect to the origin, and have meromorphic extensions to an open neighborhood of the closed unit disk. Let…

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 can be solved by proving…

A Toeplitz matrix is one in which the matrix elements are constant along diagonals. The Fisher-Hartwig matrices are much-studied singular matrices in the Toeplitz family. The matrices are defined for all orders, $N$. They are parametrized…

Let $a(z)=\sum_{i\in\mathbb Z}a_iz^i$ be a complex valued function defined for $|z|=1$, such that $\sum_{i\in\mathbb Z}|ia_i|<\infty$, and let $E=(e_{i,j})_{i,j\in\mathbb {Z}^+}$ be such that $\sum_{i,j\in\mathbb{Z}^+}|e_{i,j}|<\infty$. A…

We investigate truncated Toeplitz operators belonging to the Schatten ideals. We completely characterize such operators when they have an analytic symbol or belong to the ideal of Hilbert-Schmidt operators. We also study model spaces…

For weighted Toeplitz operators $\T^N_\phi$ defined on spaces of holomorphic functions in the unit ball, we derive regularity properties of the solutions $f$ to the integral equation $\T^N_\phi(f)=h$ in terms of the regularity of the symbol…

For a bounded function $\varphi$ on the unit circle $\mathbb T$, let $T_\varphi$ be the associated Toeplitz operator on the Hardy space $H^2$. Assume that the kernel $$K_2(\varphi):=\{f\in H^2:\,T_\varphi f=0\}$$ is nontrivial. Given a…

Consider the Toeplitz matrix $T_n(f)$ generated by the symbol $f(\theta)=\hat{f}_r e^{\mathbf{i}r\theta}+\hat{f}_0+\hat{f}_{-s} e^{-\mathbf{i}s\theta}$, where $\hat{f}_r, \hat{f}_0, \hat{f}_{-s} \in \mathbb{C}$ and $0<r<n,~0<s<n$. For…

We reconsider studies of Toeplitz operators on function spaces (the weighted Bergman space, the generalized derivative Hardy space) and the H-Toeplitz operators on the Bergman space. Past studies have considered the presence or absence of…

This work provides two results obtained as a consequence of an inversion formula for Toeplitz matrices with real symbol. First we obtain an asymptotic expression for the minimal eigenvalues of a Toeplitz matrix with a symbolwhich is…

In this paper, we study matrix representations of truncated Toeplitz operators with respect to orthonormal bases which are invariant under a canonical conjugation map. In particular, we determine necessary and sufficient conditions for when…

The low-rank matrix completion problem asks whether a given real matrix with missing values can be completed so that the resulting matrix has low rank or is close to a low-rank matrix. The completed matrix is often required to satisfy…

We characterize the completeness and frame/basis property of a union of under-sampled windowed exponentials of the form $$ {\mathcal F}(g): =\{e^{2\pi i n x}: n\ge 0\}\cup \{g(x)e^{2\pi i nx}: n<0\} $$ for $L^2[-1/2,1/2]$ by the spectra of…

Let $\lambda$ be a complex number in the closed unit disc $\overline{\Bbb D}$, and $\cal H$ be a separable Hilbert space with the orthonormal basis, say, ${\cal E}=\{e_n:n=0,1,2,\cdots\}$. A bounded operator $T$ on $\cal H$ is called a {\em…

Let $X,Y$ be Banach spaces, and fix a linear operator $T \in \mathcal{L}(X,Y)$, and ideals $\mathcal{I}, \mathcal{J}$ on $\omega$. We obtain Silverman--Toeplitz type theorems on matrices $A=(A_{n,k}: n,k \in \omega)$ of linear operators in…

Denote by $\mathcal{W}_1$ the set of complex valued functions of the form $a(z)=\sum_{i=-\infty}^{+\infty}a_iz^i$ which are continuous on the unit circle, and such that $\sum_{i=-\infty}^{+\infty}|ia_i|<\infty$. We call CQT matrix a…

In this note, we show that for hyponormal Toeplitz operators, there exists a lower bound for the area of the spectrum. This extends the known estimate for the spectral area of Toeplitz operators with an analytic symbol.

Motivated by the canonical decomposition of contractions on Hilbert spaces, we investigate when contractive Toeplitz operators on vector-valued Hardy spaces on the unit disc admit a non-zero reducing subspace on which its restriction is…

The eigenvalues of Toeplitz matrices $T_{n}(f)$ with a real-valued symbol $f$, satisfying some conditions and tracing out a simple loop over the interval $[-\pi,\pi]$, are known to admit an asymptotic expansion with the form \[…

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…