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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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