Spectral properties of Volterra-type integral operators on Fock--Sobolev spaces

Publication date: 27 February 2017

Abstract: We study some spectral properties of Volterra-type integral operators

V_{g}

and

I_{g}

with holomorphic symbol

g

on the Fock--Sobolev spaces

m a t h c a l F_{p s i_{m}}^{p}

. We showed that

V_{g}

is bounded on

m a t h c a l F_{p s i_{m}}^{p}

if and only if

g

is a complex polynomial of degree not exceeding two, while compactness of

V_{g}

is described by degree of

g

being not bigger than one. We also identified all those positive numbers

p

for which the operator

V_{g}

belongs to the Schatten

m a t h c a l S_{p}

classes. Finally, we characterize the spectrum of

V_{g}

in terms of a closed disk of radius twice the coefficient of the highest degree term in a polynomial expansion of

g

.

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