BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Brett Wick (Washington University in St. Louis)
DTSTART:20211101T190000Z
DTEND:20211101T200000Z
DTSTAMP:20260423T021339Z
UID:paw/38
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/paw/38/">Sin
 gular Integral Operators on the Fock Space</a>\nby Brett Wick (Washington 
 University in St. Louis) as part of Probability and Analysis Webinar\n\n\n
 Abstract\nIn this talk we will discuss the recent solution of a question r
 aised by K. Zhu about characterizing a class of singular integral operator
 s on the Fock space.  We show that for an entire function $\\varphi$ belon
 ging to the Fock space  ${\\mathscr F}^2(\\mathbb{C}^n)$ on the complex Eu
 clidean space $\\mathbb{C}^n$\, the integral operator\n\n\\[\nS_{\\varphi}
 F(z)=\\int_{\\mathbb{C}^n} F(w) e^{z \\cdot\\bar{w}} \\varphi(z- \\bar{w})
 \\\,d\\lambda(w)\,  \\quad z\\in\\mathbb{C}^n\,\n\\]\n\nis bounded on ${\\
 mathscr F}^2(\\mathbb{C}^n)$ if and only if there exists a function $m\\in
  L^{\\infty}(\\mathbb{R}^n)$ such that\n\n\\[\n\\varphi(z)=\\int_{\\mathbb
 {R}^n} m(x)e^{-2\\left(x-\\frac{i}{2}   z  \\right)^2} dx\, \\quad \\in\\m
 athbb{C}^n.\n\\]\nHere $d\\lambda(w)=\\pi^{-n}e^{-\\left\\vert w\\right\\v
 ert^2}dw$ is the Gaussian measure on $\\mathbb C^n$.\n\nWith this characte
 rization we are able to obtain some fundamental results of the operator $S
 _\\varphi$\, including the normality\, the $C^*$ algebraic properties\, th
 e spectrum and its compactness. Moreover\, we obtain the reducing  subspac
 es of $S_{\\varphi}$.\n\nIn particular\, in the case $n=1$\, this gives a 
 complete solution to the question proposed by K. Zhu for the Fock space ${
 \\mathscr F}^2(\\mathbb{C})$\non the complex plane ${\\mathbb C}$ (Integr.
  Equ. Oper. Theory  {\\bf  81} (2015)\, 451--454).\n\nThis talk is based o
 n joint work with Guangfu Cao\, Ji Li\, Minxing Shen\, and Lixin Yan.\n
LOCATION:https://researchseminars.org/talk/paw/38/
END:VEVENT
END:VCALENDAR