Operator theory, singular integral equations, and PDEsCopyright (c) 2023 Washington University in St. Louis All rights reserved.
https://openscholarship.wustl.edu/iwota2016/special/OTPDE
Recent Events in Operator theory, singular integral equations, and PDEsen-usWed, 06 Dec 2023 19:57:38 PST3600Notes on spectral theory on Banach spaces
https://openscholarship.wustl.edu/iwota2016/special/OTPDE/7
https://openscholarship.wustl.edu/iwota2016/special/OTPDE/7Thu, 21 Jul 2016 15:30:00 PDT
We consider a compact linear map T acting between Banach spaces both of which are uniformly convex and uniformly smooth; it is supposed that T has trivial kernel and range dense in the target space. We will try to find conditions under which the action of T is given by a series. This provides a Banach-space version of the well-known Hilbert-space result of E. Schmidt. Based on joint work/collaboration with Edmunds/Evans/Harris.
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Jan LangSpectral shift functions for differential operators.
https://openscholarship.wustl.edu/iwota2016/special/OTPDE/6
https://openscholarship.wustl.edu/iwota2016/special/OTPDE/6Thu, 21 Jul 2016 15:00:00 PDT
We will discuss extensions of the Lifshits-Krein-Koplienko spectral shift functions to cases of more general operators that include some classical differential operators.
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Anna SkripkaOperator equations and Spectral Problems
https://openscholarship.wustl.edu/iwota2016/special/OTPDE/5
https://openscholarship.wustl.edu/iwota2016/special/OTPDE/5Thu, 21 Jul 2016 14:30:00 PDT
The multiplication operators are the linear operators induced by multiplication by some fixed function. One of the important fact which motivates the study of multiplication operators is that every normal operator is similar to a multiplication operator. The another motivational fact that makes the study of multiplication operators interesting as well as demanding is its close association with various classes of operators, particularly, Toeplitz operators, Hankel operators and Composition operators. Multiplication operators have been a matter of study in operators theory over the years and these operators on different function spaces have been studied by several mathematicians. We talk about some operator equations involving these operators and discuss some spectral problems.
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Gopal DattWitten Index and spectral shift function
https://openscholarship.wustl.edu/iwota2016/special/OTPDE/4
https://openscholarship.wustl.edu/iwota2016/special/OTPDE/4Mon, 18 Jul 2016 16:00:00 PDT
Let $D$ be a selfadjoint unbounded operator on a Hilbert space and let $\{B(t)\}$ be a one parameter norm continuous family of self-adjoint bounded operators that converges in norm to asymptotes $B_\pm$. Then setting $A(t)=D+B(t)$ one can consider the operator $\mathbf{D_A}^{}=d/dt+A(t)$ on the Hilbert space $L_2(\mathbb{R},H)$. We present a connection between the theory of spectral shift function for the pair of the asymptotes $(A_+,A_-)$ and index theory for the operator $\mathbf{D_A}^{}$. Under the condition that the operator $B_+$ is a $p$-relative trace-class perturbation of $A_-$ and some additional smoothness assumption we prove a heat kernel formula for all $t>0$, $$\mathrm{tr}\Big(e^{-t\mathbf{D_A}^{}\mathbf{D_A}^{*}}-e^{-t\mathbf{D_A}^{*}\mathbf{D_A}^{}}\Big)=-\Big(\frac{t}{\pi}\Big)^{1/2}\int_0^1\mathrm{tr}\Big(e^{-tA_s^2}(A_+-A_-)\Big)ds,$$ where $A_s, s\in[0,1]$ is a straight path joining $A_-$ and $A_+$. Using this heat kernel formula we obtain the description of the Witten index of the operator $\mathbf{D_A}^{}$ in terms of the spectral shift function for the pair $(A_+,A_-)$. {\bf Theorem.} \textit{If\, $0$ is a right and a left Lebesgue point of the spectral shift function $\xi(\cdot;A_+,A_-)$ for the pair $(A_+,A_-)$ (denoted by $\xi_L(0_+; A_+,A_-)$ and $\xi_L(0_-; A_+, A_-)$, respectively), then the Witten index $W_s(\mathbf{D_A})$ of the operator $\mathbf{D_A}$ exists and equals $$W_s(\mathbf{D_A})=\frac12\big(\xi(0+;A_+,A_-)+\xi(0-;A_+,A_-)\big).$$} We note that our assumptions include the cases studied earlier. In particular, we impose no assumption on the spectra of $A_\pm$ and we can treat differential operators in any dimension. As a corollary of this theorem we have the following result. {\bf Corollary.} \textit{Assume that the asymptotes $A_\pm$ are boundedly invertible. Then the operator $\mathbf{D_A}$ is Fredholm and for the Fredholm index $\mathrm{index}(\mathbf{D_A})$ of the operator $\mathbf{D_A}$ we have $$\mathrm{index}(\mathbf{D_A})=\xi(0;A_+,A_-)=\mathrm{sf}\{A(t)\}_{t=-\infty}^{+\infty},$$ where $\mathrm{sf}\{A(t)\}_{t=-\infty}^{+\infty}$ denotes the spectral flow along the path $\{A(t)\}_{t=-\infty}^{+\infty}.$}
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Galina LevitinaMuckenhoupt Hamiltonians, triangular factorization, and Krein orthogonal entire functions
https://openscholarship.wustl.edu/iwota2016/special/OTPDE/3
https://openscholarship.wustl.edu/iwota2016/special/OTPDE/3Mon, 18 Jul 2016 15:30:00 PDT
According to classical results by M. G. Krein and L. de Branges, for every positive measure $\mu$ on the real line $\mathbb{R}$ such that $\int_{\mathbb{R}} \frac{d\mu(t)}{1 + t^2} < \infty$ there exists a Hamiltonian $H$ such that $\mu$ is the spectral measure for the corresponding canonical Hamiltonian system $JX' = z HX$. In the case where $\mu$ is an even measure from Steklov class on $\mathbb{R}$, we show that the Hamiltonian $H$ normalized by $\det H = 1$ belongs to the classical Muckenhoupt class $A_2$. Applications of this result to triangular factorizations of Wiener-Hopf operators and Krein orthogonal entire functions will be also discussed.
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Roman BessonovWeyl's Formula and Kernel Operators
https://openscholarship.wustl.edu/iwota2016/special/OTPDE/2
https://openscholarship.wustl.edu/iwota2016/special/OTPDE/2Mon, 18 Jul 2016 14:30:00 PDT
Weyl's famous formula determines the asymptotic behaviour of the eigenvalues of the Laplace Operator with Dirichlet or Neumann boundary conditions. It shows in particular that the volume is determined by the eigenvalues. In the talk we will show how the asymptotic behaviour in the spirit of Weyl is related to ultracontractivity and in particular to estimates for kernel operators. These estimates will allow us to develop a perturbation theory for Weyl asymptotics. The results are rewarding: We prove that also the Dirichlet-to-Neumann operator (in the Calderon setting with potential) has a Weyl asymptotics. This operator is defined on the L_2 space of the boundary of a smooth domain, and the main result shows that its eigenvalues determine the surface of the domain. This joint work with Tom ter Elst.
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Wolfgang ArendtOn singular integral operators with
linear-fractional involutions
https://openscholarship.wustl.edu/iwota2016/special/OTPDE/1
https://openscholarship.wustl.edu/iwota2016/special/OTPDE/1Thu, 21 Jul 2016 16:00:00 PDT
We denote the Cauchy singular integral operator along a contour $\cal L$ by $ (S_{\cal L}\varphi)(t)= \frac{1}{\pi i}\int\limits_{\cal L} \frac{\varphi(\tau)}{\tau-t}d\tau $ and the identity operator by $(I_{\cal L}\varphi)(t)=\varphi(t)$. In the paper [1,2] we constructed a similarity transformation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\begin{equation}\label{opeq T} $F^{-1}AF=D,$ \ % \vspace{-3mm} %\end{equation}\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% between the singular integral operators $A$ with the rotation operator $W_{\Bbb T}$ through the angle $2\pi /m$ on the unit circle ${\Bbb T}$, acting on the space $L_2({\Bbb T})$, and a certain matrix characteristic singular integral operator without shifts acting on the space $L_{2}^{m}(\Bbb T)$. For $m=2$, we have $(W_{\Bbb T} \varphi)(t)=\varphi(-t),$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{displaymath}%\label{aISW} A = a_{0}I_{\Bbb T} \!+\! b_{0}S_{\Bbb T}\! +\! a_{1}W_{\Bbb T}\! +\! b_{1}S_{\Bbb T}W_{\Bbb T},\quad A\in [L_2(\Bbb T)],D=uI_{\Bbb T}+vS_{{\Bbb T}},\quad D\in [L_2^2(\Bbb T)]. \end{displaymath} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\[ %D=uI_{\Bbb T}+vS_{{\Bbb T}},\quad D\in [L_2^2(\Bbb T)] %\] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In the same papers acting by invertible operators from the right hand and left-hand side we reduced \begin{displaymath}%\label{aISQ} B_{\Bbb R}=aI_{\Bbb R}+bQ_{\Bbb R}+cS_{\Bbb R}+dQ_{\Bbb R}S_{\Bbb R},\ B_{\Bbb R}\in[L_2({\Bbb R})],\ {\Bbb R}=(+\infty,-\infty), \end{displaymath} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% where involution $ \left(Q_{\Bbb R}\varphi\right)(x)= \frac{\sqrt{\delta^2+\beta}}{x-\delta}\varphi[\alpha(x)], \ \alpha(x)=\frac{\delta x + \beta}{x-\delta}, \ \delta^2+\beta>0$,\\ to a matrix characteristic singular integral operator without shift: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % $\mathcal{H}A\mathcal{E}$, \[ {\mathcal{H}}B{\mathcal{F}}=D_{{\Bbb R}_+}, \quad D_{{\Bbb R}_+}=u{\Bbb R}_+I_{{\Bbb R}_+}+v{\Bbb R}_+S_{{\Bbb R}_+}, \] acting on the space $L_2^{2}({\Bbb R}_+,x^{-\frac{1}{4}}),\ {{\Bbb R}_+}= (0,+\infty) $. We will refer to the formulas as operator equalities. Different applications of operator equalities to singular integral operators and to boundary value problems are considered. \medskip %Operators equalities are main tools \cite{Ka01 SMM}, \cite{Ka07 %MathNachr}. %\begin{thebibliography}{99} %\bibitem{Ka01 SMM} [1] A. A. Karelin, On a relation between singular integral operators with a Carleman linear-fractional shift and matrix characteristic operators without s hift, Boletin Soc. Mat. Mexicana Vol. 7 No. 12 (2001), pp. 235--246. %\bibitem{Ka07 MathNachr} [2] A. Karelin, Aplications of operator equalities to singular integral operators and to Riemann boundary value problems, Math. Nachr. Vol. 280 No. 9-10 (2007), pp. 1108--1117.
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Oleksandr Karelin