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On a Product Formula for Unitary Groups
Author(s) -
Cachia V.
Publication year - 2005
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609305004479
Subject(s) - mathematics , hilbert space , semigroup , unitary state , subsequence , separable space , pure mathematics , exponential function , product (mathematics) , degenerate energy levels , combinatorics , discrete mathematics , mathematical analysis , bounded function , political science , law , physics , geometry , quantum mechanics
For any nonnegative self‐adjoint operators A and B in a separable Hilbert space, the Trotter‐type formula[ ( e i 2 t A / n + e i 2 t B / n) / 2 ] nis shown to converge strongly in the norm closure of dom ( A 1/2 ) ∩ dom ( B 1/2 for some subsequence and for almost every t ∈ R. This result extends to the degenerate case, and to Kato‐functions following the method of T. Kato (see ‘Trotter's product formula for an arbitrary pair of self‐adjoint contraction semigroup’, Topics in functional analysis (ed. M. Kac, Academic Press, New York, 1978) 185–195). Moreover, the restrictions on the convergence can be removed by considering functions other than the exponential. 2000 Mathematics Subject Classification 47D03 (primary), 47B25 (secondary).