The Schwinger cocycle on algebras with unbounded operators

In this article, we describe a class of algebras with unbounded operators on which the Schwinger cocycle extends. For this, we replace a space of bounded operators commonly used in the literature by some space of (maybe unbounded) tame operators, in particular by spaces of pseudo-differential operators, acting on the space of sections of a vector bundle E→M. We study some particular examples which we hope interesting or instructive. The case of classical and log-polyhomogeneous pseudo-differential operators is studied, because it carries other cocycles, defined with renormalized traces of pseudo-differential operators, that are some generalizations of the Khesin-Kravchenko-Radul cocycle. The present construction furnishes a simple proof of an expected result: The cohomology class of these cocycles are the same as cohomology class of the Schwinger cocycle. When M=S¹, we show that the Schwinger cocycle is non-trivial on many algebras of pseudo-differential operators (these operators need not to be classical or bounded). These two results complete the work and extend the results of a previous work [J.-P. Magnot, Renormalized traces and cocycles on the algebra of S¹-pseudo-differential operators, Lett. Math. Phys. 75 (2) (2006) 111-127]. When dim(M)>1, we furnish a new example of sign operator which could suggest that the framework of pseudo-differential operators is not adapted to all the cases. On this example, we have to work on some algebras of tame operators, in order to show that the Schwinger cocycle has a non-vanishing cohomology class.     

Perturbations and expressions for generalized inverses in Banach spaces and Moore-Penrose inverses in Hilbert spaces of closed linear operators

The problems of perturbation and expression for the generalized inverses of closed linear operators in Banach spaces and for the Moore-Penrose inverses of closed linear operators in Hilbert spaces are studied. We first provide some stability characterizations of generalized inverses of closed linear operators under T-bounded perturbation in Banach spaces, which are exactly equivalent to that the generalized inverse of the perturbed operator has the simplest expression T⁺(I+δTT⁺)^-1. Utilizing these results, we investigate the expression for the Moore-Penrose inverse of the perturbed operator in Hilbert spaces and provide a unified approach to deal with the range preserving or null space preserving perturbation. An explicit representation for the Moore-Penrose inverse of the perturbation is also given. Moreover, we give an equivalent condition for the Moore-Penrose inverse to have the simplest expression T^†(I+δTT^†)^-1. The results obtained in this paper extend and improve many recent results in this area.     

The ultimate estimate of the upper norm bound for the summation of operators

Let A and B be bounded linear operators acting on a Hilbert space H. It is shown that the triangular inequality serves as the ultimate estimate of the upper norm bound for the sum of two operators in the sense that

sup{∥U^*AU+V^*BV∥:U and V are unitaries}=min{∥A+μI∥+∥B-μI∥:μ∈C}.     

Solution of nonlinear integral equations of Hammerstein type

Let E be a 2-uniformly real Banach space and F,K:E→E be nonlinear-bounded accretive operators. Assume that the Hammerstein equation u+KFu=0 has a solution. A new explicit iteration sequence is introduced and strong convergence of the sequence to a solution of the Hammerstein equation is proved. The operators F and K are not required to satisfy the so-called range condition. No invertibility assumption is imposed on the operator K and F is not restricted to be an angle-bounded (necessarily linear) operator.     

Lifting strong commutants of unbounded subnormal operators

Various theorems on lifting strong commutants of unbounded subnormal (as well as formally subnormal) operators are proved. It is shown that the strong symmetric commutant of a closed symmetric operatorS lifts to the strong commutant of some tight selfadjoint extension ofS. Strong symmetric commutants of orthogonal sums of subnormal operators are investigated. Examples of (unbounded) irreducible subnormals, pure subnormals with rich strong symmetric commutants and cyclic subnormals with highly nontrivial strong commutants are discussed.This work was supported by the KBN grant # 2P03A 041 10.     

Reflexive *-derivations and lattices of invariant subspaces of operator algebras associated with them

The paper studies unbounded reflexive *-derivations δ of C*-algebras of bounded operators on Hilbert spaces H whose domains D(δ) are weekly dense in B(H and contain compact operators. It describes a one-to-one correspondence between these derivations and pairs S,L, where S are symmetric densely operators on H and L are J-orthogonal π-reflexive lattices of subspaces in the deficiency spaces of S. The domains D(δ) of these *-derivations are associated with some non-selfadjoint reflexive algebras A_δ of bounded operators on H⊕H. The paper analyzes the structure of the lattices of invariant subspaces of A_δ and of the normalizers of A_δ-the largest Lie subalgebras of B(H⊕H) such that A_δ are their Lie ideals.     

On simultaneous triangularization of collections of compact operators

In this paper we consider collections of compact (resp. C_p class) operators on arbitrary Banach (resp. Hilbert) spaces. For a subring R of reals, it is proved that an R-algebra of compact operators with spectra in R on an arbitrary Banach space is triangularizable if and only if every member of the algebra is triangularizable. It is proved that every triangularizability result on certain collections, e.g., semigroups, of compact operators on a complex Banach (resp. Hilbert) space gives rise to its counterpart on a real Banach (resp. Hilbert) space. We use our main results to present new proofs as well as extensions of certain classical theorems (e.g., those due to Kolchin, McCoy, and others) on arbitrary Banach (resp. Hilbert) spaces.     

Toeplitz operators on the Fock space: Radial component effects

The paper is devoted to the study of specific properties of Toeplitz operators with (unbounded, in general) radial symbolsa=a(r). Boundedness and compactness conditions, as well as examples, are given. It turns out that there exist non-zero symbols which generate zero Toeplitz operators. We characterize such symbols, as well as the class of symbols for whichT _a =0 impliesa(r)=0 a.e. For each compact setM there exists a Toeplitz operatorT _a such that spT _a =ess-spT _a =M. We show that the set of symbols which generate bounded Toeplitz operators no longer forms an algebra under pointwise multiplication.Besides the algebra of Toeplitz operators we consider the algebra of Weyl pseudodifferential operators obtained from Toeplitz ones by means of the Bargmann transform. Rewriting our Toeplitz and Weyl pseudodifferential operators in terms of the Wick symbols we come to their spectral decompositions.This work was partially supported by CONACYT Project 27934-E, México.The first author acknowledges the RFFI Grant 98-01-01023, Russia.     

Functional calculus for generators of uniformly bounded holomorphic semigroups

We present a method for constructing a functional calculus for (possibly unbounded) operators that generate a uniformly bounded holomorphic semigroup, e^−zA. (A will be called a generator.) These are closed, densely defined operators whose spectrum and numerical range are contained in [0,∞), with respect to an equivalent norm.     

Unbounded order continuous operators on Riesz spaces

In this paper, using the concept of unbounded order convergence in Riesz spaces, we define new classes of operators, named unbounded order continuous (uo-continuous, for short) and boundedly unbounded order continuous operators. We give some conditions under which uo-continuity will be equivalent to order continuity of some operators on Riesz spaces. We show that the collection of all uo-continuous linear functionals on a Riesz space E is a band of \(E^\sim \).     

Isometric Equivalence of Certain Operators on Banach Spaces

A pair of operators on a Banach space X are isometrically equivalent if they are intertwined by a surjective isometry of X. We investigate the isometric equivalence problem for pairs of operators on specific types of Banach spaces. We study weighted shifts on symmetric sequence spaces, elementary operators acting on an ideal I of Hilbert space operators, and composition operators on the Bloch space. This last case requires an extension of known results about surjective isometries of the Bloch space.     

Unbounded pseudodifferential calculus on Lie groupoids

We develop an abstract theory of unbounded longitudinal pseudodifferential calculus on smooth groupoids (also called Lie groupoids) with compact basis. We analyze these operators as unbounded operators acting on Hilbert modules over C^∗(G), and we show in particular that elliptic operators are regular. We construct a scale of Sobolev modules which are the abstract analogues of the ordinary Sobolev spaces, and analyze their properties. Furthermore, we show that complex powers of positive elliptic pseudodifferential operators are still pseudodifferential operators in a generalized sense.     

Spectral inclusion for unbounded block operator matrices

In this paper we establish a new analytic enclosure for the spectrum of unbounded linear operators A admitting a block operator matrix representation. For diagonally dominant and off-diagonally dominant block operator matrices, we show that the recently introduced quadratic numerical range W²(A) contains the eigenvalues of A and that the approximate point spectrum of A is contained in the closure of W²(A). This provides a new method to enclose the spectrum of unbounded block operator matrices by means of the non-convex set W²(A). Several examples illustrate that this spectral inclusion may be considerably tighter than the one by the usual numerical range or by perturbation theorems, both in the non-self-adjoint case and in the self-adjoint case. Applications to Dirac operators and to two-channel Hamiltonians are given.     

Anti-self-dual Lagrangians: Variational resolutions of non-self-adjoint equations and dissipative evolutions

We develop the concept and the calculus of anti-self-dual (ASD) Lagrangians and their derived vector fields which seem inherent to many partial differential equations and evolutionary systems. They are natural extensions of gradients of convex functions – hence of self-adjoint positive operators – which usually drive dissipative systems, but also provide representations for the superposition of such gradients with skew-symmetric operators which normally generate unitary flows. They yield variational formulations and resolutions for large classes of non-potential boundary value problems and initial-value parabolic equations. Solutions are minima of newly devised energy functionals, however, and just like the self (and anti-self) dual equations of quantum field theory (e.g. Yang–Mills) the equations associated to such minima are not derived from the fact they are critical points of the functional I, but because they are also zeroes of suitably derived Lagrangians. The approach has many advantages: it solves variationally many equations and systems that cannot be obtained as Euler–Lagrange equations of action functionals, since they can involve non-self-adjoint or other non-potential operators; it also associates variational principles to variational inequalities, and to various dissipative initial-value first order parabolic problems. These equations can therefore be analyzed with the full range of methods – computational or not – that are available for variational settings. Most remarkable are the permanence properties that ASD Lagrangians possess making their calculus relatively manageable and their domain of applications quite broad.     

Chaotic unbounded differentiation operators

We construct dense sets of hypercyclic vectors for unbounded differention operators, including differentiation operators on the Hardy spaceH ₂, and the Laplacian operator onL ₂((), for any bounded open subset of ². Furthermore, we show that these operators are chaotic, in the sense of Devaney.     

Decomposition of a scalar operator into a product of unitary operators with two points in spectrum

We consider products of unitary operators with at most two points in their spectra, 1 and e^iα. We prove that the scalar operator e^iγI is a product of k such operators if α(1+1/(k-3))?γ?α(k-1-1/(k-3)) for k?5. Also we prove that for e^iα≠-1, only a countable number of scalar operators can be decomposed in a product of four operators from the mentioned class. As a corollary we show that every unitary operator on an infinite-dimensional space is a product of finitely many such operators.     

On Some Product of Two Unbounded Self-Adjoint Operators

We give a spectral analysis of some unbounded normal product HK of two self-adjoint operators H and K (which appeared in [7]) and we say why it is not self-adjoint even if the spectrum of one of the operators is sufficiently “asymmetric”. Then, we investigate the self-adjointness of KH (given it is normal) for arbitrary self-adjoint H and K by giving a counterexample and some positive results and hence finishing off with the whole question of normal products of self-adjoint operators (appearing in [1, 7, 12]). The author was supported in part by CNEPRU: B01820070020 (Ministry of Higher Education, Algeria).