A generalized Riesz-Schauder decomposition theorem

Let M be a von Neumann algebra of operators on a Hilbert space H, τ be a faithful normal semifinite trace on M. We define two (closed in the topology of convergence in measure τ) classes P1 and P2 of τ-measurable operators and investigate their properties. The class P2 contains P1. If a τ-measurable operator T is hyponormal, then T lies in P1; if an operator T lies in Pk, then UTU* belongs to Pk for all isometries U from M and k = 1, 2; if an operator T from P1 admits the bounded inverse T−1, then T−1 lies in P1. We establish some new inequalities for rearrangements of operators from P1. If a τ-measurable operator T is hyponormal and Tn is τ-compact for some natural number n, then T is both normal and τ-compact. If M = B(H) and τ = tr, then the class P1 coincides with the set of all paranormal operators on H.

A quasi-expectation of a C ∗ {C^ \ast } -algebra A on a C ∗ {C^\ast } -subalgebra B is a bounded linear projection Q : A → B Q:A \to B such that Q ( x a y ) = x Q ( a ) y ∀ x , y ∈ B , a ∈ A Q(xay) = xQ(a)y \forall x,y \in B,a \in A . It is shown that if M is a von Neumann algebra of operators on Hilbert space H for which there exists a quasi-expectation of B ( H ) B(H) on M, then there exists a projection of norm one of B ( H ) B(H) on M, i.e. M is injective. Further, if M is an amenable von Neumann subalgebra of a von Neumann algebra N, then there exists a quasi-expectation of N on M ′ ∩ N M’ \cap N . These two facts yield as an immediate corollary the recent result of A. Connes that all amenable von Neumann algebras are injective.

We consider the Schur–Horn problem for normal operators in von Neumann algebras, which is the problem of characterizing the possible diagonal values of a given normal operator based on its spectral data. For normal matrices, this problem is well known to be extremely difficult, and in fact, it remains open for matrices of size greater than 3 . We show that the infinite-dimensional version of this problem is more tractable, and establish approximate solutions for normal operators in von Neumann factors of type I∞, II, and III. A key result is an approximation theorem that can be seen as an approximate multivariate analogue of Kadison's Carpenter Theorem.

Let $\mathcal{M}$ be a semifinite von Neumann algebra on a Hilbert space $\mathcal{H}$ equipped with a faithful normal semifinite trace $\tau$, $S(\mathcal{M},\tau)$ be the ${}^*$-algebra of all $\tau$-measurable operators. Let $S_0(\mathcal{M},\tau)$ be the ${}^*$-algebra of all $\tau$-compact operators and $T(\mathcal{M},\tau)=S_0(\mathcal{M},\tau)+\mathbb{C}I$ be the ${}^*$-algebra of all operators $X=A+\lambda I$ with $A\in S_0(\mathcal{M},\tau)$ and $\lambda \in \mathbb{C}$. It is proved that every operator of $T(\mathcal{M},\tau)$ that is left-invertible in $T(\mathcal{M},\tau)$ is in fact invertible in $T(\mathcal{M},\tau)$. It is a generalization of Sterling Berberian theorem (1982) on the subalgebra of thin operators in $\mathcal{B} (\mathcal{H})$. For the singular value function $\mu(t; Q)$ of $Q=Q^2\in S(\mathcal{M},\tau)$, the inclusion $\mu(t; Q)\in \{0\}\bigcup [1, +\infty)$ holds for all $t>0$. It gives the positive answer to the question posed by Daniyar Mushtari in 2010.

The spectral properties of a compact operator \(T : X \longrightarrow Y\) on a normed linear space resemble those of square matrices. For a compact operator, the spectral properties can be treated fairly completely in the sense that Fredholm's famous theory of integral equations may be extended to linear functional equations \(T x -\lambda\) \(= y\) with a complex parameter \(\lambda\) . This paper has studied and investigated the spectral properties of compact operators in Hilbert spaces. The spectral properties of compact linear operators are relatively simple generalization of the eigenvalues of finite matrices. As a result, the paper has given a number of corresponding propositions and interesting facts which are used to prove basic properties of compact operators. The Fredholm theory has been introduced to investigate the solvability of linear integral equations involving compact operators.

From the viewpoint of the theory of orthomodular lattices of elementary propositions, Quantum Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Sol\'er's theorem. The said lattice eventually coincides with the lattice of all orthogonal projectors on a separable Hilbert space over R, C, or over the algebra of quaternions H. Quantum states are $\sigma$-additive probability measures on that non-Boolean lattice. Gleason's theorem proves that, if the Hilbert space is separable with dimension >2 and the Hilbert space is either real or complex, then states are one-to-one with standard density matrices (self-adjoint, positive, unit-trace, trace-class operators). The extension of this result to quaternionic Hilbert spaces was obtained by Varadarajan in 1968. Unfortunately, even if the hard part of the proof is correct, the formulation of this extension is mathematically incorrect. This is due to some peculiarities of the notion of trace in quaternionic Hilbert spaces, e.g., basis dependence, making the theory of trace-class operators in quaternionic Hilbert spaces different from the standard theory in real and complex Hilbert spaces. A minor issue also affects Varadarajan's statement for real Hilbert space formulation. This paper is mainly devoted to present Gleason-Varadarajan's theorem into a technically correct form valid for the three types of Hilbert spaces. After having develped part of the general mathematical technology of trace-class operators in (generally non-separable) quaternionic Hilbert spaces, we prove that only the {\em real part} of the trace enters the formalism of quantum theories (also dealing with unbounded observables and symmetries) and it can be safely used to formulate and prove a common statement of Gleason's theorem.

In order to make this monograph self-contained, we summarize in this chapter some basic definitions and results for unbounded linear operators in a Hilbert space. In Section 1.1, we recall the definitions of C*-algebras and von Neumann algebras. In Section 1.2, we define and investigate the notion of closedness, the closure and the adjoint of an unbounded linear operator in a Hilbert space. Section 1.3 is devoted to the Cayley transform approach to the self-adjointness of a symmetric operator. Section 1.4 deals with the self-adjoint extendability of a symmetric operator with help of the deficiency spaces. In Section 1.5, we extend to unbounded self-adjoint operators the spectral theorem and the functional calculus theorem for bounded self-adjoint operators. Section 1.6 is devoted to Stone’s theorem. In Section 1.7, the polar decomposition of bounded linear operators is extended to closed linear operators. Section 1.8 introduces Nelson’s analytic vector theorem for the selfadjointness of closed symmetric operators. Section 1.9, finally, deals with the form representation theorem and the Friedrichs self-adjoint extension theorem.

Let T be a compact linear operator acting in a complex Hilbert space H. Then there is a simple resolution of the identity {Eλ} in H which reduces T (for a proof of this statement, and for definitions of the terms used, see (l), section 5, especially Theorem 6). In the present paper we give constructions for the resolvent of T, and for a complete set of principal (that is, eigen and adjoined) vectors, in terms of T, {Eλ}, and the diagonal coefficients {αλ} of T. These constructions require no technique more involved than the expansion of a resolvent operator in a Neumann series.

Variational problem on solutions of the free phase problem an compact operator in Hilbert spaces is considered. Local relaxation property of the gradient method and weak convergence are proven.

In the first part of this paper we present a representation theorem for the directional derivative of the metric projection operator in an arbitrary Hilbert space. As a consequence of the representation theorem, we present in the second part the development of the theory of projected dynamical systems in infinite dimensional Hilbert space. We show that this development is possible if we use the viable solutions of differential inclusions. We use also pseudomonotone operators.

Remark on the decay for damped string and beam equations

It is an old problem in operator theory whether a pair of norm one compact Hermitian operators with "small" (in norm) commutator can be "well" approximated by a commuting pair of Hermitian operators.We show that, for operators of rank not exceeding n, such approximants exist provided ||[A, B]||/n 1/2 is small.This improves a result of Pearcy and Shields and sheds some new light on the original question and its relationship to a few related ones.

The multiplicities of the powers of a bounded linear operator T T , acting on a complex separable infinite-dimensional Banach space X \mathfrak {X} , satisfy the inequalities \[ ( ∗ ∗ ) μ ( T n ) ⩽ μ ( T h n ) ⩽ h μ ( T n ) for all h , n ⩾ 1. ( * * )\qquad \mu ({T^n}) \leqslant \mu ({T^{hn}}) \leqslant h\mu ({T^n})\quad {\text {for}}\;{\text {all}}\;h,n \geqslant 1. \] Nothing else can be said, in general, because simple examples show that for each sequence { μ n } n = 1 ∞ \{ {\mu _n}\} _{n = 1}^\infty , satisfying the inequalities ( ∗ ∗ ) ( * * ) , there exists T T acting on X \mathfrak {X} such that μ ( T n ) = μ n \mu ({T^n}) = {\mu _n} for all n ⩾ 1 n \geqslant 1 .

An effective method of investigation of positive maps on the set of positive definite operators

As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.