*-algebras of unbounded operators affiliated with a von Neumann algebra

In the paper, the *-algebras of measurable operators, locally measurable operators, and τ-measurable operators associated with a von Neumann algebra M are considered. Conditions under which some of these algebras coincide are given. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 183–197.     

An application of the Fourier method of separation of variables to constructing exactly solvable deformations of partial differential operators

As is known, in mathematical physics there are differential operators with constant coefficients whose fundamental solutions can be constructed explicitly; such operators are said to be exactly solvable. In this paper, the problem of adding lower-order terms with variable coefficients to exactly solvable operators in such a way that the new operators (deformations) admit constructing fundamental solutions in explicit form is posed. This problem is directly related to Hadamard’s problem of describing differential operators satisfying the Huygens’ principle. On the basis of the Fourier method of separation of variables and the method of gauge-equivalent operators, an effective method for finding exactly solvable deformations depending on one variable is constructed. An application of such deformations to constructing Huygens’ differential operators associated with the cone of real symmetric positive-definite matrices is suggested. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

Equal-time temperature correlators of the one-dimensional heisenberg XY chain

For equal-time temperature correlators of the anisotropic Heisenberg XY chain, representations are obtained in the form of determinants of M×M matrices. These representations are simple deformations of the answers for the isotropic XXO chain. In the thermodynamic limit, the correlators are expressed in terms of the Fredholm determinants of linear integral operators. Bibliography: 30 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 245, 1997, pp. 173–206. Translated by N. A. Kitanin.     

Deforming the Lie superalgebra of contact vector fields on <Emphasis Type="Italic">S</Emphasis><Superscript>1|2</Superscript> inside the Lie superalgebra of pseudodifferential operators on <Emphasis Type="Italic">S</Emphasis><Superscript>1|2</Superscript>

We classify deformations of the standard embedding of the Lie superalgebra $ \mathcal{K} $ \mathcal{K} (2) of contact vector fields on the (1, 2)-dimensional supercircle into the Lie superalgebra SΨD(S ^1|2 ) of pseudodifferential operators on the supercircle S ^1|2 . The proposed approach leads to the deformations of the central charge induced on $ \mathcal{K} $ \mathcal{K} (2) by the canonical central extension of SΨD(S ^1|2 ).     

Parabolic Marcinkiewicz integrals along surfaces on product domains

In this paper, parabolic Marcinkiewicz integral operators along surfaces on the product domain ℝ ⁿ × ℝ ^m (n,m ⩾ 2) are introduced. L ^p bounds of such operators are obtained under weak conditions on the kernels.     

Spectral order and isotonic differential operators of Laguerre–Pólya type

The spectral order on R ⁿ induces a natural partial ordering on the manifold of monic hyperbolic polynomials of degree n. We show that all differential operators of Laguerre–Pólya type preserve the spectral order. We also establish a global monotony property for infinite families of deformations of these operators parametrized by the space ℓ^∞ of real bounded sequences. As a consequence, we deduce that the monoid of linear operators that preserve averages of zero sets and hyperbolicity consists only of differential operators of Laguerre–Pólya type which are both extensive and isotonic. In particular, these results imply that any hyperbolic polynomial is the global minimum of its -orbit and that Appell polynomials are characterized by a global minimum property with respect to the spectral order.     

A remark on the range of elementary operators

Let L(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Given A ∈ L(H), we define the elementary operator Δ _A : L(H) → L(H) by Δ _A (X) = AXA − X. In this paper we study the class of operators A ∈ L(H) which have the following property: ATA = T implies AT*A = T* for all trace class operators T ∈ C ₁(H). Such operators are termed generalized quasi-adjoints. The main result is the equivalence between this character and the fact that the ultraweak closure of the range of Δ _A is closed under taking adjoints. We give a characterization and some basic results concerning generalized quasi-adjoints operators.     

Pseudodifferential Operators on Periodic Graphs

The main aim of the paper is Fredholm properties of a class of bounded linear operators acting on weighted Lebesgue spaces on an infinite metric graph Γ which is periodic with respect to the action of the group \mathbb Zⁿ{{\mathbb {Z}}^n} . The operators under consideration are distinguished by their local behavior: they act as (Fourier) pseudodifferential operators in the class OPS ⁰ on every open edge of the graph, and they can be represented as a matrix Mellin pseudodifferential operator on a neighborhood of every vertex of Γ. We apply these results to study the Fredholm property of a class of singular integral operators and of certain locally compact operators on graphs.     

Generalized indices of operators inB(H)

A generalized dimension is further developed. Here subtraction and addition of two generalized dimensions are defined, so that the operations: ∞ ± n = ∞, ∞ + ∞ = ∞, which used to play an inflexible role, are refined and moreover, ∞ - ∞, which used to be meaningless, is done in sense. Then generalized index for semi-Fred-holm operators is developed to wholeB(H), i.e. all of bounded linear operators in Hilbert spaceH. Theorem 2.2 is proved with an example, which is in contradiction to a known proposition for semi-Fredholm operators in form, practically a refined result of the known proposition. Then, it is proved thatB(H) is the union of countably many disjoint arewise connected sets over all the generalized dimensions ofB(H). Project supported by the National Natural Science Foundation of China     

On the stability of the index of unbounded nonlocal operators in Sobolev spaces

Unbounded operators corresponding to nonlocal elliptic problems on a bounded region G ⊂ ℝ² are considered. The domain of these operators consists of functions in the Sobolev space W ₂ ^m (G) that are generalized solutions of the corresponding elliptic equation of order 2m with the right-hand side in L ₂(G) and satisfy homogeneous nonlocal boundary conditions. It is known that such unbounded operators have the Fredholm property. It is proved that lower order terms in the differential equation do not affect the index of the operator. Conditions under which nonlocal perturbations on the boundary do not change the index are also formulated. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 255, pp. 116–135.     

Iteration operators on a set of continuous functions of Baire space

Continuous functions on Baire space are considered. Iteration operators are defined on a set of continuous functions. The idea of a module of continuity of a function is introduced. The condition for the growth of module of continuity φ whose satisfaction guarantees that for any enumerable sequence of integration operators and any natural n there exists (n + 1) argument function with the module of continuity φ which cannot be obtained from n-argument functions with the module of continuity φ using any operator of this sequence is formulated. Examples of iteration operators are given.     

The contact boundary of a complex polynomial

 We define the contact boundary of a complex polynomial f : ℂ ⁿ → ℂ as the intersection of some generic fiber with a large sphere. We show that, up to contact isotopy, this does not depend on the choice of the fiber (provided it is generic) and is invariant under polynomial automorphism of ℂ ⁿ . We next prove that the formal homotopy class of this contact boundary is invariant in a large family of deformations of polynomials, which are not necessarily topologically trivial. Received: 15 November 2002 Published online: 20 March 2003 Mathematics Subject Classification (2000): 32S55, 53D15, 32S50     

Differential and Analytical Properties of Semigroups of Operators

We systematically analyze differential and analytical properties of various kinds of semigroups of linear operators, including (local) convoluted C-semigroups and ultradistribution semigroups. The study of differentiable integrated semigroups leans heavily on the unification of the approaches of Barbu (Ann Scuola Norm Sup Pisa 23:413–429, 1969) and Pazy (Semigroups of linear operators and applications to partial differential equations. Springer, Berlin, 1983). We furnish illustrative examples of operators which generate differentiable integrated semigroups, further analyze the analytic properties of solutions of the backwards heat equation, and prove that several introduced classes of differentiable semigroups persist under bounded ‘commuting’ perturbations.     

Weighted norm inequalities,off-diagonal estimates and elliptic operators Part II: Off-diagonal estimates on spaces of homogeneous type

This is the second part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. We consider a substitute to the notion of pointwise bounds for kernels of operators which usually is a measure of decay. This substitute is that of off-diagonal estimates expressed in terms of local and scale invariant L^p − L^q estimates. We propose a definition in spaces of homogeneous type that is stable under composition. It is particularly well suited to semigroups. We study the case of semigroups generated by elliptic operators. This work was partially supported by the European Union (IHP Network “Harmonic Analysis and Related Problems” 2002-2006, Contract HPRN-CT-2001-00273-HARP). The second author was also supported by MEC “Programa Ramón y Cajal, 2005” and by MEC Grant MTM2004-00678.     

On Diagonal Subdifferential Operators in Nonreflexive Banach Spaces

Consider a real-valued bifunction f defined on C ×C, where C is a closed and convex subset of a Banach space X, which is concave in its first argument and convex in its second one. We study its subdifferential with respect to the second argument, evaluated at pairs of the form (x,x), and the subdifferential of − f with respect to its first argument, evaluated at the same pairs. We prove that if f vanishes whenever both arguments coincide, these operators are maximal monotone, under rather undemanding continuity assumptions on f. We also establish similar results under related assumptions on f, e.g. monotonicity and convexity in the second argument. These results were known for the case in which the Banach space is reflexive and C = X. Here we use a different approach, based upon a recently established sufficient condition for maximal monotonicity of operators, in order to cover the nonreflexive and constrained case (C ≠ X). Our results have consequences in terms of the reformulation of equilibrium problems as variational inequality ones.     

Essential Spectral Inclusions for Operators on Banach Spaces

This paper centers on local spectral conditions that are both necessary and sufficient for the equality of the essential spectra of two bounded linear operators on complex Banach spaces that are intertwined by a pair of bounded linear mappings. In particular, if the operators T and S are intertwined by a pair of injective operators, then S is Fredholm provided that T is Fredholm and S has property (δ) in a neighborhood of 0. In this case, ind(T) ≤ ind(S), and equality holds precisely when the eigenvalues of the adjoint T* do not cluster at 0. By duality, we obtain refinements of results due to Putinar, Takahashi, and Yang concerning operators with Bishop’s property (β) intertwined by pairs of operators with dense range. Moreover, we establish an extension of a result due to Eschmeier that, under appropriate assumptions regarding the single-valued extension property, leads to necessary and sufficient conditions for quasi-similar operators to have equal essential spectra. In particular it turns out that the single-valued extension property plays an essential role in the preservation of the index in this context.      

Algebraic axiomatization of tense intuitionistic logic

We introduce two unary operators G and H on a relatively pseudocomplemented lattice which form an algebraic axiomatization of the tense quantifiers “it is always going to be the case that” and “it has always been the case that”. Their axiomatization is an extended version for the classical logic and it is in accordance with these operators on many-valued Łukasiewicz logic. Finally, we get a general construction of these tense operators on complete relatively pseudocomplemented lattice which is a power lattice via the so-called frame.     

Interpolation of operators of weak type (ϕ<Subscript>0</Subscript>, ψ<Subscript>0</Subscript>, ϕ<Subscript>1</Subscript>, ψ<Subscript>1</Subscript>) in Lorentz spaces

We prove theorems on interpolation of quasilinear operators of weak type (ϕ₀, ψ₀, ϕ₀, ψ₁) in Lorentz spaces. The operators under study are analogs of the Calderón operator and the Benett operator for concave and convex functions ϕ₀(t), ψ₀(t), ϕ₁(t), and ψ₁(t). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1490–1507, November, 2005.     

Gellerstedt and Laplace–Beltrami operators relative to a mixed signature metric

Gellerstedt and Laplace–Beltrami operators relative to a certain mixed signature metric share among themselves an interesting and important property: under suitable change of coordinates they can be represented, up to a multiplying factor, in terms of Q _α and R _α singular first order perturbations of the Laplace and D’Alembertian operators. Knowledge of fundamental solutions for Q _α and R _α leads us to finding explicit formulas for fundamental solutions to those operators. J. Barros-Neto’s research partially supported by NSF, Grant # INT 0124940. F. Cardoso’s research partially supported by CNPq (Brazil).     

On the infimum problem of Hilbert space effects

The quantum effects for a physical system can be described by the set ε(H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator I. The infimum problem of Hilbert space effects is to find under what condition the infimum A∧B exists for two quantum effects A and B∈ε(H). The problem has been studied in different contexts by R. Kadison, S. Gudder, M. Moreland, and T. Ando. In this note, using the method of the spectral theory of operators, we give a complete answer of the infimum problem. The characterizations of the existence of infimum A∧B for two effects A. B∈ε(H) are established.