$C_0$-semigroups of 2-isometries and Dirichlet spaces

Abstract: In the context of a theorem of Richter, we establish a similarity between

C_{0}

-semigroups of analytic 2-isometries

{T (t)}_{t g e q 0}

acting on a Hilbert space

m a t h c a l H

and the multiplication operator semigroup

{M_{p h i_{t}}}_{t g e q 0}

induced by

p h i_{t} (s) = e x p (- s t)

for

s

in the right-half plane

m a t h b b C_{+}

acting boundedly on weighted Dirichlet spaces on

m a t h b b C_{+}

. As a consequence, we derive a connection with the right shift semigroup

{S_{t}}_{t g e q 0}

S_tf(x)=left { �egin{array}{ll} 0 & mbox { if }0leq tleq x, \ f(x-t)& mbox { if } x>t, end{array} ight .

acting on a weighted Lebesgue space on the half line

m a t h b b R_{+}

and address some applications regarding the study of the invariant subspaces of

C_{0}

-semigroups of analytic 2-isometries.

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