BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Maria Nowak (Maria Curie-Skłodowska University\, Poland)
DTSTART:20201006T130000Z
DTEND:20201006T140000Z
DTSTAMP:20260422T201824Z
UID:CAvid/15
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CAvid/15/">O
 n kernels of Toeplitz operators</a>\nby Maria Nowak (Maria Curie-Skłodows
 ka University\, Poland) as part of CAvid: Complex Analysis video seminar\n
 \nLecture held in N/A.\n\nAbstract\nLet $H^2$ denote the standard Hardy sp
 ace on the unit disk $\\mathbb\nD$ and let $\\mathbb T=\\partial \\mathbb 
 D$. Every $f(z)=\\sum_{n=0}^{\\infty}a_nz^n\\in H^2$ has a nontangential l
 imit $f(e^{i\\theta})$ a.e. on $\\mathbb {T}=\\partial\\mathbb {D}$ and th
 is boundary function  $f(e^{i\\theta})$ is in $L^2(\\mathbb {T})$.\nFurthe
 rmore\, if $\\{c_n\\}$ are Fourier coefficients of $f(e^{i\\theta})$ then 
 $c_n=a_n$ for $n\\geq 0$ and $c_n=0$ for $n<0$.\n Actually\, the space $H^
 2$  can be identified with a closed subspace of\n$L^2(\\mathbb {T})$  whos
 e Fourier coefficients with negative indices vanish.\n\n\nFor $\\varphi\\i
 n\nL^{\\infty}(\\mathbb T)$ the Toeplitz operator $T_{\\varphi}$ on $H^2$ 
 is given by\n$T_{\\varphi}f=P_{+}(\\varphi f)$\, where $P_{+}$ is the orth
 ogonal\nprojection of $L^2(\\mathbb T)$ onto $H^2$.  It is a consequence o
 f Hitt's Theorem  that\n$\\ker T_{\\varphi}= fK_I$\, where $K_I= H^2\\omin
 us IH^2$\nis the model space corresponding to the inner function $I$ such 
 that\n$I(0)=0$ and $f$ is an outer function of unit $H^2$ norm that\nacts 
 as an isometric multiplier from  $K_I$ onto $f K_{I}$.\nHowever\,  not all
   spaces $fK_{I}$\, where $f$ and $K_I$ are  as above\,  can be  kernels o
 f Toeplitz operators.\nThe sufficient and necessary condition for the spac
 e $fK_I$ to be the kernel of a Toeplitz operator was given by E. Hayashi (
 1990).\nIn 1994 D. Sarason gave another proof of this condition based on d
 e Branges-Rovnyak  spaces theory.\nIf $M= fK_I$ is a kernel of a Toeplitz 
 operator\, then also we have $M=\\ker T_{\\frac{\\overline{If}}{f}}$\nIn t
 he talk we consider the case when $fK_I\\varsubsetneq \\ker T_{\\frac{\\ov
 erline{If}}{f}}$ and try to describe\nthe space $\\ker T_{\\frac{\\bar I\\
 bar f}{f}}\\ominus fK_I$. We use  Sarason's approach and   the structure  
 of de Branges-Rovnyak  spaces generated by nonextreme functions.\n\nThe ta
 lk is based on joint work with P. Sobolewski\, A.\nSo{\\l}tysiak and M. Wo
 {\\l}oszkiewicz-Cyll.\n
LOCATION:https://researchseminars.org/talk/CAvid/15/
END:VEVENT
END:VCALENDAR