On Certain Automorphisms of Sets of Partial Isometries

Under the mild condition of continuity at a single point we describe all the bijections of the set of all partial isometries on a Hilbert space which preserve the order and the orthogonality in both directions. Moreover, we present a natural analogue of Wigner’s theorem on quantum mechanical symmetries for the set of all rank-1 partial isometries. In the mathematical foundations of quantum mechanics the lattice P (H) of all projections on a Hilbert space H plays a fundamental role. The whole set of projections equipped with the usual partial ordering and orthogonality represents the probabilistic aspect of the theory and the set P1(H) of all rank-1 projections with the notion of transition probability is the object of the fundamental theorem of Wigner on quantum mechanical symmetries. Clearly, the projections can be characterized as positive partial isometries. If we drop positivity, one can naturally raise several problems concerning partial isometries which are familiar in relation to projections. Probably the most fundamental results concerning the mentioned structures of projections are the description of all bijective transformations of P (H) which preserve the order and the orthogonality in both directions, and Wigner’s theorem (sometimes called unitary-antiunitary theorem) determining all bijective transformations of P1(H) which preserve the transition probabilities (these results can be found, for example, in [1]; also see the references therein). The aim of this paper is to present analogue results for the corresponding structures of partial isometries. These operators, we mean the partial isometries, play the same role in the relatively new theory of ternary structures of operators as the projections do in connection with the usual binary structures. The former structures are of great importance in several subareas of functional analysis, for example, in the study of the geometrical properties of operator algebras (see [2] and its references). We begin with the notation that we use throughout. Let H be a complex separable Hilbert space. The algebra of all bounded linear operators on H is denoted by B(H). The symbol F (H) stands for the set of all finite rank elements of B(H). An operator U ∈ B(H) is called a partial isometry if Date: February 1, 2008. 1991 Mathematics Subject Classification. Primary: 47B49. This research was supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. T030082, T031995, and by the Ministry of Education, Hungary, Reg. No. FKFP 0349/2000. 1