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J. A. Virtanen

Publications and source records attributed to J. A. Virtanen.

9 recordsLinked to original sources
arXiv2025

Volterra-type inner derivations on Hardy spaces

A classical result of Calkin [Ann. of Math. (2) 42 (1941), pp. 839-873] says that an inner derivation $S\mapsto [T,S] = TS-ST$ maps the algebra of bounded operators on a Hilbert space into the ideal of compact operators if and only if $T$ is a compact perturbation of the multiplication by a scalar. In general, an analogous statement fails for operators on Banach spaces. To complement Calkin's result, we characterize Volterra-type inner derivations on Hardy spaces using generalized area operators and compact intertwining relations for Volterra and composition operators. Further, we characterize the compact intertwining relations for multiplication and composition operators between Hardy and Bergman spaces.

math.FA↗
arXiv2024

The Hilbert matrix done right

We give very simple proofs of the classical results of Magnus and Hill on the spectral properties of the Hilbert matrix $$ H = \left ( {1 \over i+j+ 1 } \right )_{i,j\geq 0} $$ which defines a bounded linear operator on the sequence space $\ell^2$. In particular, we use the Mehler-Fock transform to find the spectrum and the latent eigenfunctions of the Hilbert matrix, that is, we show that the spectrum of $H$ is $[0,π]$ with no eigenvalues (Magnus' result) and describe all complex sequences $x$ such that $Hx=μx$ for some complex number $μ$ (Hill's result).

math.FA↗
arXiv2024

A survey of asymptotics of determinants for structured matrices

In this survey we show how to produce asymptotics of determinants of structured matrices using operator theory methods. We describe the asymptotics for finite Toeplitz matrices, finite Toeplitz plus Hankel matrices and generalizations of these, and also finite sections of functions of Toeplitz operators, all with smooth matrix-valued symbols. Many of these results are well-known. However, the results for certain Toeplitz plus Hankel operators with matrix-valued symbol are presented for the first time and the result for the finite sections of functions of Toeplitz operators is new.

math.FA↗
arXiv2024

Asymptotics of block Toeplitz determinants with piecewise continuous symbols

We determine the asymptotics of the block Toeplitz determinants $\det T_n(ϕ)$ as $n\to\infty$ for $N\times N$ matrix-valued piecewise continuous functions $ϕ$ with a finitely many jumps under mild additional conditions. In particular, we prove that $$ \det T_n(ϕ) \sim G^n n^ΩE\quad {\rm as}\ n\to \infty, $$ where $G$, $E$, and $Ω$ are constants that depend on the matrix symbol $ϕ$ and are described in our main results. Our approach is based on a new localization theorem for Toeplitz determinants, a new method of computing the Fredholm index of Toeplitz operators with piecewise continuous matrix-valued symbols, and other operator theoretic methods. As an application of our results, we consider piecewise continuous symbols that arise in the study of entanglement entropy in quantum spin chain models.

math.FA↗
arXiv2024

Generalized Hilbert matrix operators acting on Bergman spaces

In this article we study the generalized Hilbert matrix operator $Γ_μ$ acting on the Bergman spaces $A^p$ of the unit disc for $1\leq p<\infty$. In particular, we characterize the measures $μ$ for which the operator $Γ_μ$ is bounded and we provide estimates of its operator norm. Finally, we also describe when $Γ_μ$ is compact by computing its essential norm.

math.CV↗
arXiv2023

Essential positivity

We define essentially positive operators on Hilbert space as a class of self-adjoint operators whose essential spectra is contained in the nonnegative real numbers and describe their basic properties. Using Toeplitz operators and the Berezin transform, we further illustrate the notion of essential positivity in the Hardy space and the Bergman space.

math.FA↗
arXiv2022

Generalized Volterra type integral operators on large Bergman spaces

Let $ϕ$ be an analytic self-map of the open unit disk $\mathbb{D}$ and $g$ analytic in $\mathbb{D}$. We characterize boundedness and compactness of generalized Volterra type integral operators $$GI_{(ϕ,g)}f(z)= \int_{0}^{z}f'(ϕ(ξ))\,g(ξ)\, dξ$$ and $$GV_{ (ϕ, g)}f(z)= \int_{0}^{z} f(ϕ(ξ))\,g(ξ)\, dξ, $$ acting between large Bergman spaces $A^p_ω$ and $A^q_ω$ for $0<p,q\le \infty$. To prove our characterizations, which involve Berezin type integral transforms, we use the Littlewood-Paley formula of Constantin and Peláez and establish corresponding embedding theorems, which are also of independent interest. When $ϕ(z) = z$, our results for $GV_{(ϕ,g)}$ complement the descriptions of Pau and Peláez.

math.CV↗
arXiv2020

Entanglement entropy of two disjoint intervals separated by one spin in a chain of free fermion

We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, \texttt{arXiv:1104.1004}, for the reduced density of states of two disjoint intervals with lattice sites $P=\{1,2,\dots,m\}\cup\{2m+1,2m+2,\dots, 3m\}$, which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher-Hartwig symbol using the the Riemann--Hilbert method.

math-ph↗