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Trajectory spaces, generalized functions and unbounded operators. (English) Zbl 0622.46032

Lecture Notes in Mathematics, 1162, Berlin etc.: Springer-Verlag. IV, 272 p. DM 38.50 (1985).
This book presents recent work due to the authors. Its purpose is to construct dual couples (S,T) of locally convex topological vector spaces S and T as in the case that S is a space of test functions and \(T=S'\) (dual of S) is the corresponding space of generalized functions. Its method relies on operator theory in Hilbert spaces. Let A be a nonnegative self-adjoint (unbounded) operator in a separable Hilbert space X. For every \(t>0\), the operator \(e^{-tA}\) is bounded with dense range \(X_ t=e^{-tA}(X)\) and inverse \(e^{tA}\). \(X_ t\) gets its own Hilbert space structure such that the one-to-one mapping \(e^{-tA}:\) \(X\to X_ t\) be unitary. Let \(S_{X,A}\) be the union of the spaces \(X_ t\), \(t>0\), equipped with the inductive limit topology induced by the Hilbert spaces \(X_ t\). \(S_{X,A}\) is called the analyticitity space. A trajectory of the semigroup \((e^{-tA})_{t>0}\) is a mapping F: (0,\(\infty)\to X\) such that \(F(t+s)=e^{-sA}F(t)\), s, \(t>0\). For all \(t>0\), \(F(t)\in S_{X,A}\). The trajectory space \(T_{X,A}\) is the sense of all such F equipped with the locally convex topology induced by the family of norms \(\| F(t)\|_ X\), \(t>0\). An element x of X is identified with the trajectory \(t\mapsto e^{-tA}x\); so one gets an embedding \(X\subset T_{X,A}\), X dense subspace of \(T_{X,A}\). Given \(u\in S_{X,A}\) and \(F\in T_{X,A}\) the inner product \((e^{tA}u\), F(t)) in X does not depend on \(t>0\) (t small enough so that \(u\in e^{- tA}X)\). The sesquilinear form \(<u,F>=(e^{tA}u\), F(t)) is a pairing: each one of the spaces \(S_{X,A}\) and \(T_{X,A}\) may be identified with the dual of the other. \(S_{X,A}\subset X\subset T_{X,A}\) is called a Gelfand triple.
Chapters 1, 3 and 4 present the functional analytic theory of such triples: linear topological properties, continuous linear mappings, nuclearity, tensor products, kernel theorems, operator algebras. In Chapter 2 some classical examples of Gelfand triples are discussed: here X is an \(L^ 2\)-space and A a differential operator. Chapter 5 gives a mathematical interpretation of Dirac’s formalism in quantum mechanics. Bra and ket vectors are considered as elements of suitable trajectory spaces. A more detailed exposition of that matter may be found in the book ”A mathematical introduction to Dirac’s formalism” by the same authors [North-Holland Math. Library, 36, (1986; review below)].
Reviewer: P.Jeanquartier

MSC:

46E99 Linear function spaces and their duals
47D03 Groups and semigroups of linear operators
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46A20 Duality theory for topological vector spaces

Citations:

Zbl 0622.46033
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