Hilbert–Schmidt operator

Abstract: Let A be a proper H*-algebra and let r(A) be the set of all products xy of members x, y of A. Then r(A) is a normed algebra with respect to some norm r( ) which is related to the norm I || of A by the equality: IIalI2=r(a*a), aCA. There is a trace tr defined on r(A) such that tr(a) = (ae,, ea) for each aEr(A) and each maximal family {e,,} of mutually orthogonal projections in A. The trace is related to the scalar product of A by the equality: tr(xy) = (x, y*) = (y, x*) for all x, y EA. 1. The trace-class of operators (Trc) was introduced by R. Schatten [5] as the set of all products of Hilbert-Schmidt operators acting on a Hilbert space. This class has its own norm, in which it is complete, and a trace, which can be used to define a scalar product on the set (oc) of all Hilbert-Schmidt operators to convert it into a simple H*-algebra. The present work deals with a generalization of this theory to an arbitrary H*-algebra. There are two ways this generalization can be achieved. One way would be to decompose an H*-algebra into simple H*-algebras, represent each simple H*-algebra as a Hilbert-Schmidt class of operators, construct the corresponding trace-classes, take their direct sum and then derive the desired properties of the class 'r(A) = { xy I x, ycA } through identifying it with this direct sum. The other approach, subject of this paper, is to apply Schatten's technique directly to an H*-algebra. 2. Let A be a proper H*-algebra (A is a Banach algebra whose norm is a Hilbert space norm and which has an involution x-x* such that (y, x*z) = (xy, z) = (x, zy*) for all x, y, z in A (see [1])). A right centralizer on A is a bounded operator S on A such that (Sx)y = S(xy) for all x, yEA [2]. Note that each operator of the form La:x--ax (x, aEA) is a right centralizer on A. A projection in A is a nonzero member e of A such that e2 =e=e* -O (e is a nonzero selfadjoint Presented to the Society, February 25, 1967; received by the editors August 15, 1968 and, in revised form, October 17, 1969. AMS 1969 subject classifications. Primary 4650, 4660; Secondary 4615.

Abstract: We introduce the symmetric approximation of frames by normalized tight frames extending the concept of the symmetric orthogonalization of bases by orthonormal bases in Hilbert spaces. We prove existence and uniqueness results for the symmetric approximation of frames by normalized tight frames. Even in the case of the symmetric orthogonalization of bases, our techniques and results are new. A crucial role is played by whether or not a certain operator related to the initial frame or basis is Hilbert-Schmidt.

Abstract: Viability and invariance problems related to a stochastic equation in a Hilbert space H are studied. Finite dimensional invariant C2 submanifolds of H are characterized. We derive Nagumo type conditions and prove a regularity result: Any weak solution, which is viable in a finite dimensional C2 submanifold, is a strong solution. These results are related to finding finite dimensional realizations for stochastic equations. There has recently been increased interest in connection with a model for the stochastic evolution of forward rate curves.

Abstract: We consider a model Schrodinger operator Hμ associated with a system of three particles on the threedimensional lattice ℤ3 with a functional parameter of special form. We prove that if the corresponding Friedrichs model has a zero-energy resonance, then the operator Hμ has infinitely many negative eigenvalues accumulating at zero (the Efimov effect). We obtain the asymptotic expression for the number of eigenvalues of Hμ below z as z → −0.