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An extension of Berezin’s approximation method. (English) Zbl 0817.47033

Let \(B\) be a normal operator in a Hilbert space \(K\) with domain \(D(B)\) whose spectrum \(\sigma (B)\) is included in a sector \(1+ S_ \theta \equiv\{ 1+z\); \(z\in \mathbb{C}\), \(|\arg z|< \theta\}\), \(0\leq \theta< \pi/2\). Let \(A\) be the restriction of \(B\) to a closed subspace \(H\) of \(K\): \(Af= PBf\), \(f\in D(A)= H\cap D(B)\), where \(P\) is the orthogonal projection of \(K\) onto \(H\). Assume that \(H\cap D(B)\) is dense in \(H\). Then \(A\) is a densely defined, closable operator in \(H\). \(| B_ 2|\) denotes the absolute value of the imaginary part \(B_ 2\) of \(B= B_ 1+ iB_ 2\). The paper under review proves that if \(P| B_ 2| P\) is bounded on \(H\) and \((2\cos \theta- 1)\cos \theta> \sin \theta\), then there exists a closed extension \(\widetilde {A}\) of \(A\) which generates a \(C_ 0\)-semigroup and for every fixed \(t>0\) the sequence \(\{( Pe^{-(t/n)B} P)^ n \}\) is strongly convergent to \(\exp (-t \widetilde {A})\) on \(H\) as \(n\to \infty\). This generalizes F. A. Berezin’s result for \(B\) a selfadjoint operator in [“Covariant and contravariant symbols of operators”, Izv. Akad. Nauk SSSR, Ser. Mat. 36, 1134-1167 (1972; Zbl 0247.47019)]. An application is given to Toeplitz operators in the Bargmann-Segal space.

MSC:

47B25 Linear symmetric and selfadjoint operators (unbounded)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces

Citations:

Zbl 0247.47019
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