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A semigroup of operators in the boson Fock space. (English. Russian original) Zbl 0738.47035
Funct. Anal. Appl. 24, No. 2, 135-144 (1990); translation from Funkts. Anal. Prilozh. 24, No. 2, 63-73 (1990).
Let $$F(V_ n)$$ be a space of holomorphic functions, where $$V_ n$$ is complex Hilbert space of the dimension $$n=1,2,\dots,\infty$$. The scalar product is $$\langle f,g\rangle=\iint f(z)\overline {g(z)}\exp(-(z,z))dz d\bar z$$. The author investigates the problem of the boundedness of the operators $Bf(z)=\iint\exp\left\{ {1\over 2} (z\bar u)\begin{pmatrix} K &L \\ M &N \end{pmatrix} {z^ t \choose \bar u^ t} \right\} f(u)\exp(-(u,u))du d\bar u.$ These operators represent a semigroup of generalized linear- fractional Krein’s transformations of the matrix ball of infinite dimension. The connection between these operators and representation theory is investigated.

MSC:
 47D06 One-parameter semigroups and linear evolution equations 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 46E20 Hilbert spaces of continuous, differentiable or analytic functions
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References:
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