## Operator theory, singular integral equations, and PDEs

#### Location

Cupples I Room 215

#### Start Date

7-21-2016 4:00 PM

#### End Date

21-7-2016 4:20 PM

#### Description

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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %$$\label{opeq T} F^{-1}AF=D, \ % \vspace{-3mm} %$$\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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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Jul 21st, 4:00 PM Jul 21st, 4:20 PM

On singular integral operators with linear-fractional involutions

Cupples I Room 215

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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %$$\label{opeq T} F^{-1}AF=D, \ % \vspace{-3mm} %$$\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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.