Christopher Hammond, Jennifer Moorhouse, Marian E. Robbins
Building on techniques developed by Cowen and Gallardo-Gutiérrez, we find a concrete formula for the adjoint of a composition operator with rational symbol acting on the Hardy space $H^{2}$. We consider some specific examples, comparing our formula with several results that were previously known.
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Norm bounds for self-adjoint Toeplitz operators with non-radial symbols on the Fock space
本研究将Galbis的Fock空间自伴Toeplitz算子范数界推广至非径向有界可积符号，通过移植Nicola-Tilli等周不等式与适配Galbis的二维积分逼近论证完成拓展。
This paper contains a further analysis of the Toeplitz-like operators $T_ω$ on $H^p$ with rational symbol $ω$ having poles on the unit circle that were previously studied in [5.6]. Here the adjoint operator $T_ω^*$ is described. In the case where $p=2$ and $ω$ has poles only on the unit circle $\mathbb{T}$, a description is given for when $T_ω^*$ is symmetric and when $T_ω^*$ admits a selfadjoint extension. Also in the case where $p=2$, $ω$ has only poles on $\mathbb{T}$ and in addition $ω$ is proper, it is shown that $T_ω^*$ coincides with the unbounded Toeplitz operator defined by Sarason in…
We give an elementary proof of a formula recently obtained by Hammond, Moorhouse, and Robbins for the adjoint of a rationally induced composition operator on the Hardy space H^2. We discuss some variants and implications of this formula, and use it to provide a sufficient condition for a rationally induced composition operator adjoint to be a compact perturbation of a weighted composition operator.
In this paper a definition is given for an unbounded Toeplitz-like operator with rational symbol which has poles on the unit circle. It is shown that the operator is Fredholm if and only if the symbol has no zeroes on the unit circle, and a formula for the index is given as well. Finally, a matrix representation of the operator is discussed.
Unitary rational functions: The scaled quaternion case
本文在缩放四元数环上发展有理函数的最小实现与酉分解理论，利用非经典伴随和实希尔伯特空间性质，并初步建立底层环的矩阵理论。
A Toeplitz-like operator with rational matrix symbol having poles on the unit circle:\ Matrix representation and spectral analysis
本文利用状态空间实现技术研究符号在单位圆上有极点的无界Toeplitz算子，建立其与半无限Toeplitz矩阵的联系，并确定本质谱与预解集。
This paper is a continuation of our study of a class of Toeplitz-like operators with a rational symbol which has a pole on the unit circle. A description of the spectrum and its various parts, i.e., point, residual and continuous spectrum, is given, as well as a description of the essential spectrum. In this case, the essential spectrum need not be connected in ${\mathbb C}$. Various examples illustrate the results.
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…
This paper concerns the analysis of an unbounded Toeplitz-like operator generated by a rational matrix function having poles on the unit circle T. It extends the analysis of such operators generated by scalar rational functions with poles on T found in [11,12,13]. A Wiener-Hopf type factorization of rational matrix functions with poles and zeroes on T is proved and then used to analyze the Fredholm properties of such Toeplitz-like operators. A formula for the index, based on the factorization, is given. Furthermore, it is shown that the determinant of the matrix function having no zeroes on T …
Schatten class membership of products of composition operators and their adjoints
本文研究Hardy与Bergman空间上复合算子及其伴随算子乘积的Schatten类成员性，给出相关算子属于Schatten类的充要条件及特定符号下的充要条件。
Building on techniques used in the case of the disc, we use a variety of methods to develop formulae for the adjoints of composition operators on Hardy spaces of the upper half-plane. In doing so, we prove a slight extension of a known necessary condition for the boundedness of such operators, and use it to provide a complete classification of the bounded composition operators with rational symbol. We then consider some specific examples, comparing our formulae with each other, and with other easily deduced formulae for simple cases.
Determinant formulas for finite Toeplitz plus Hankel matrices with rational symbols
本文将Day公式推广至有理符号的有限Toeplitz加Hankel矩阵行列式，通过Fredholm行列式给出精确表达式，并用于分析极限特征值。
A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Invertibility and Riccati equations
针对单位圆上含极点的有理矩阵符号所诱导的无界类 Toeplitz 算子，本文用状态空间方法将其可逆性归结为相关非对称离散代数 Riccati 方程的稳定解，并给出伪典范分解与逆算子公式。
On the symbol calculus for multidimensional Hausdorff operators
该研究围绕多维Hausdorff算子的符号演算展开，基于正规Hausdorff算子对角化方法，分自伴交换扰动矩阵、附加正定假设两种情形推导$L^2(\mathbb{R}^n)$上的相关演算规则。
Let $R$ be a rational function of degree at least two, let $J_R$ be the Julia set of $R$ and let $μ^L$ be the Lyubich measure of $R$. We study the C$^*$-algebra $\mathcal{MC}_R$ generated by all multiplication operators by continuous functions in $C(J_R)$ and the composition operator $C_R$ induced by $R$ on $L^2(J_R, μ^L)$. We show that the C$^*$-algebra $\mathcal{MC}_R$ is isomorphic to the C$^*$-algebra $\mathcal{O}_R (J_R)$ associated with the complex dynamical system $\{R^{\circ n} \}_{n=1} ^\infty$.
We describe large classes of compact self-adjoint Hankel operators whose eigenvalues have power asymptotics and obtain explicit expressions for the coefficient in front of the leading term. The results are stated both in the discrete and continuous representations for Hankel operators. We also elucidate two key principles underpinning the proof of such asymptotic relations. We call them {\it the localization principle} and {\it the symmetry principle}. The localization principle says that disjoint components of the singular support of the symbol of a Hankel operator make independent contributi…
Self-Adjointness of Toeplitz Operators on the Segal-Bargmann Space
本文提出了保证无界算子值符号Toeplitz算子自伴性的新准则，适用于Lipschitz连续导数符号，通过推广Berger-Coburn估计至向量值Segal-Bargmann空间，并应用于Schrödinger表示中二次型的自伴性证明。
A simple criterion for essential self-adjointness of Weyl pseudodifferential operators
本文提出一个不依赖椭圆型假设的Weyl伪微分算子本质自伴性新判据，适用于无限维Hilbert空间上的厄米算子值符号，基于相空间微分演算、Calderón-Vaillancourt定理及Toeplitz算子自伴性结果。
We realize norms of most composition operators acting on the Hardy space with linear fractional symbol as roots of hypergeometric functions. This realization leads to simple necessary and sufficient conditions on the symbol to exhibit extremal non-compactness, establishes equivalence of cohyponormality and cosubnormality of composition operators with linear fractional symbol, and yields a complete classification of those linear fractional that induce composition operators whose norms are determined by the action of the adjoint on the normalized reproducing kernels in the Hardy space.
In this paper, we introduce and study a class of pseudo-differential operators on the lattice $\mathbb{Z}^n$. More preciously, we consider a weighted symbol class $M_{ρ, Λ}^m(\mathbb{ Z}^n\times \mathbb{T}^n), m\in \mathbb{R}$ associated to a suitable weight function $Λ$ on $\mathbb{ Z }^n$. We study elements of the symbolic calculus for pseudo-differential operators associated with $M_{ρ, Λ}^m(\mathbb{ Z}^n\times \mathbb{T}^n)$ by deriving formulae for the composition, adjoint, transpose. We define the notion of $M$-ellipticity for symbols belonging to $M_{ρ, Λ}^m(\mathbb{ Z}^n\times \mathbb{…
The characteristic functions of multivariate Feller processes with generator of affine type, and with smooth symbol functions have an explicit representation in terms of power series with rational number coefficients and with monmoms consisting of powers of the the symbol functions and formal derivatives of the symbol functions. The power series repesentations are convergent globally in time and on bounded domains of arbitrary size. Generalized symbol functions can be derived leading to power series expansions which are convergent on arbitrary domains in special cases. The rational number coef…
For self-adjoint operators $A, B$, a bounded operator $J$, and a function $f:\mathbb R\to\mathbb C$ we obtain bounds in quasi-normed ideals of compact operators for the difference $f(A)J-Jf(B)$ in terms of the operator $AJ-JB$. The focus is on functions $f$ that are smooth everywhere except for finitely many points. A typical example is the function $f(t) = |t|^γ$ with $γ\in (0, 1)$. The obtained results are applied to derive a two-term quasi-classical asymptotic formula for the trace $tr f(S)$ with $S$ being a Wiener-Hopf operator with a discontinuous symbol.
A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Fredholm characteristics
本文研究单位圆上含极点的有理矩阵符号所生成无界类 Toeplitz 算子的核维数，在 Wiener–Hopf 型分解的附加条件下给出公式，并推广相关核刻画。
In this paper we develop the global symbolic calculus of pseudo-differential operators generated by a boundary value problem for a given (not necessarily self-adjoint or elliptic) differential operator. For this, we also establish elements of a non-self-adjoint distribution theory and the corresponding biorthogonal Fourier analysis. We give applications of the developed analysis to obtain a-priori estimates for solutions of operators that are elliptic within the constructed calculus.
Given a separable unital C*algebra $C$, let $E_n$ denote the Hilbert module equal to the completion of the Schwartz space of rapidly decreasing smooth functions from $R^n$ to $C$ equipped with the $C$-valued inner product given by integration. Let $B$ denote the space of all smooth functions with bounded derivatives from $R^n$ to $C$. For each $a$ in $B$, let $O(a)$ denote the pseudodifferential operator of symbol $a$. $O$ maps $B$ to $H$, the set of all adjointable operators on $E_n$ which have smooth orbit under the canonical action of the Heisenberg group. We construct a left inverse for $O…
Self-adjoint Toeplitz operators have purely absolutely continuous spectrum. For Toeplitz operators $T$ with piecewise continuous symbols, we suggest a further spectral classification determined by propagation properties of the operator $T$, that is, by the behavior of $\exp(-iTt) f$ for $t\to\pm\infty$. It turns out that the spectrum is naturally partitioned into three disjoint subsets: thick, thin and mixed spectra. On the thick spectrum, the propagation properties are modeled by the continuous part of the symbol, whereas on the thin spectrum, the model operator is determined by the jumps of …
Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. Relying on a basis of pseudodifferential projections commuting with $A$, we construct an almost-unitary pseudodifferential operator that diagonalizes $A$ modulo an infinitely smoothing operator. We provide an invariant algorithm for the computation of its full symbol, as well as an explicit closed formula for its subprincipal symbol. Finally, we give a quantitative description of the relation between th…
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