The boundedness and the fredholm property of integral operators with anisotropically homogeneous kernels of compact type and variable coefficients

In the space L _p (? ⁿ ), 1 < p < ??, we study a new wide class of integral operators with anisotropically homogeneous kernels. We obtain sufficient conditions for the boundedness of operators from this class. We consider the Banach algebra generated by operators with anisotropically homogeneous kernels of compact type and multiplicatively slowly oscillating coefficients. We establish a relationship between this algebra and multidimensional convolution operators, and construct a symbolic calculus for it. We also obtain necessary and sufficient conditions for the Fredholm property of operators from this algebra.     

On the Solvability of Integral Operators with Bihomogeneous Kernels of the Compact Type and Variable Coefficients

In this paper, we consider the C ^? -algebra Ω_mult(? ⁿ ) of multiplicatively weakly oscillating functions and a new wide class of kernels of compact type, which includes the class of SO(n)-invariant kernels. For the Banach algebra generated by operators with kernels of this class and coefficients from Ω_mult(? ⁿ ), we construct a symbolic calculus, obtain necessary and sufficient conditions for the presence of the Fredholm property, and propose a method of calculating the index of families. Similar results are obtained for operators with bihomogeneous kernels of compact type and multiplicatively weakly oscillating coefficients, i.e., for operators from the tensor product \( {{\mathfrak{M}}_{p,n}}_{{_1}}\otimes {{\mathfrak{M}}_{p,n}}_{{_2}} \) .     

On the constant and step in Jackson’s inequality for best approximations by trigonometric polynomials and by Haar polynomials

We consider the C*-algebra generated by multidimensional integral operators with (?n)th-order homogeneous kernels and by the operators of multiplication by oscillating coefficients of the form |x|^iα. For this algebra, we construct an operator symbolic calculus and obtain necessary and sufficient conditions for the Fredholm property of an operator in terms of this calculus.     

Volterra type integral operators with homogeneous kernels in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We consider multidimensional integral Volterra type operators with kernels homogeneous of degree (?n); the operators act in L _p -spaces with a submultiplicative weight. For these operators we obtain necessary and sufficient conditions of their invertibility. Besides, we describe the Banach algebra generated by the operators. For this algebra we construct the symbolic calculus, in terms of which we obtain an invertibility criterion of the operators.     

On some notions of spectra for quaternionic operators and for n-tuples of operators

The S-spectrum has been introduced for the definition of the S-functional calculus that includes both the quaternionic functional calculus and a calculus for n-tuples of nonnecessarily commuting operators. The notion of right spectrum for right linear quaternionic operators has been widely used in the literature, especially in the context of quaternionic quantum mechanics. Moreover, several results in linear algebra, like the spectral theorem for quaternionic matrices, involve the right spectrum. In this Note we prove that the two notions of S-spectrum and of right spectrum coincide.     

Differential calculus for linear operators represented by finite signed measures and applications

We introduce a differential calculus for linear operators represented by a family of finite signed measures. Such a calculus is based on the notions of g-derived operators and processes and g-integrating measures, g?being a right-continuous nondecreasing function. Depending on the choice of?g, this differential calculus works for non-smooth functions and under weak integrability conditions. For linear operators represented by stochastic processes, we provide a characterization criterion of g-differentiability in terms of characteristic functions of the random variables involved. Various illustrative examples are considered. As an application, we obtain an efficient algorithm to compute the Riemann zeta function ??(z) with a geometric rate of convergence which improves exponentially as ?(z) increases.     

Operator pencils on the algebra of densities

We continue to study equivariant pencil liftings and differential operators on the algebra of densities. We emphasize the role played by the geometry of the extended manifold where the algebra of densities is a special class of functions. Firstly we consider basic examples. We give a projective line of diff(M)-equivariant pencil liftings for first order operators and describe the canonical second order self-adjoint lifting. Secondly we study pencil liftings equivariant with respect to volume preserving transformations. This helps to understand the role of self-adjointness for the canonical pencils. Then we introduce the Duval-Lecomte-Ovsienko (DLO) pencil lifting which is derived from the full symbol calculus of projective quantisation. We use the DLO pencil lifting to describe all regular proj-equivariant pencil liftings. In particular, the comparison of these pencils with the canonical pencil for second order operators leads to objects related to the Schwarzian.     

Area integral functions for sectorial operators on L spaces

Area integral functions are introduced for sectorial operators on L^p-spaces. We establish the equivalence between the square and area integral functions for sectorial operators on L^p spaces. This follows that the results of Cowling, Doust, McIntosh, Yagi, and Le Merdy on H^infin functional calculus of sectorial operators on L^p-spaces hold true when the square functions are replaced by the area integral functions.     

Berezin-Toeplitz quantization and composition formulas

Extending results in [L.A. Coburn, The measure algebra of the Heisenberg group, J. Funct. Anal. 161 (1999) 509-525; L.A. Coburn, On the Berezin-Toeplitz calculus, Proc. Amer. Math. Soc. 129 (11) (2001) 3331-3338] we derive composition formulas for Berezin-Toeplitz operators with i.g. unbounded symbols in the range of certain integral transforms. The question whether a finite product of Berezin-Toeplitz operators is an operator of this type again can be answered affirmatively in several cases, but there are also well-known counter examples. We explain some consequences of such formulas to C^∗-algebras generated by Toeplitz operators.     

Analyse de l'espace des phases et calcul pseudo-differential sur le groupe de Heisenberg

We establish pseudo-differential calculus on the Heisenberg group by defining an algebra of operators acting continuously on Sobolev spaces and containing the class of differential operators. Our approach puts into light microlocal directions and completes, with the Littlewood–Paley theory developed by Bahouri et al., a microlocal analysis of the Heisenberg group. To cite this article: H. Bahouri et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Area Integral Functions and H Functional Calculus for Sectorial Operators on Hilbert Spaces

Area integral functions are introduced for sectorial operators on Hilbert spaces. We establish the equivalence relationship between the square and area integral functions. This immediately extends McIntosh/Yagi's results on H^∞ functional calculus of sectorial operators on Hilbert spaces to the case when the square functions are replaced by the area integral functions.     

The algebra of Calderón-Zygmund kernels on a homogeneous group is inverse-closed

We show that the subalgebra of convolution operators with Calderón-Zygmund kernels on a homogeneous group G is inverse-closed in the algebra of all bounded linear operators on the Hilbert space L ²(G). The main tool used is a symbolic calculus, where the convolution of distributions on the group is translated via the abelian Fourier transform into a “twisted product” of symbols on the dual to the Lie algebra g of G.     

A new functional calculus for noncommuting operators

In this paper we use the notion of slice monogenic functions [F. Colombo, I. Sabadini, D.C. Struppa, Slice monogenic functions, Israel J. Math., in press] to define a new functional calculus for an n-tuple T of not necessarily commuting operators. This calculus is different from the one discussed in [B. Jefferies, Spectral Properties of Noncommuting Operators, Lecture Notes in Math., vol. 1843, Springer-Verlag, Berlin, 2004] and it allows the explicit construction of the eigenvalue equation for the n-tuple T based on a new notion of spectrum for T. Our functional calculus is consistent with the Riesz-Dunford calculus in the case of a single operator.     

Gauss-Mean Value Formula for Triharmonic Functions and its Applications in Clifford Analysis

In this paper, the integral representation for some polyharmonic functions with values in a universal Clifford algebra Cl(V_n,n) is studied and Gauss-mean value formula for triharmonic functions with values in a Clifford algebra Cl(V_n,n) are proved by using Stokes formula and higher order Cauchy-Pompeiu formula. As application some results about growth condition at infinity are obtained.     

Noncommutative residue for Heisenberg manifolds. Applications in CR and contact geometry

This paper has four main parts. In the first part, we construct a noncommutative residue for the hypoelliptic calculus on Heisenberg manifolds, that is, for the class of ΨHDO operators introduced by Beals-Greiner and Taylor. This noncommutative residue appears as the residual trace on integer order ΨHDOs induced by the analytic extension of the usual trace to non-integer order ΨHDOs. Moreover, it agrees with the integral of the density defined by the logarithmic singularity of the Schwartz kernel of the corresponding ΨHDO. In addition, we show that this noncommutative residue provides us with the unique trace up to constant multiple on the algebra of integer order ΨHDOs. In the second part, we give some analytic applications of this construction concerning zeta functions of hypoelliptic operators, logarithmic metric estimates for Green kernels of hypoelliptic operators, and the extension of the Dixmier trace to the whole algebra of integer order ΨHDOs. In the third part, we present examples of computations of noncommutative residues of some powers of the horizontal sublaplacian and the contact Laplacian on contact manifolds. In the fourth part, we present two applications in CR geometry. First, we give some examples of geometric computations of noncommutative residues of some powers of the horizontal sublaplacian and of the Kohn Laplacian. Second, we make use of the framework of noncommutative geometry and of our noncommutative residue to define lower-dimensional volumes in pseudohermitian geometry, e.g., we can give sense to the area of any 3-dimensional CR manifold endowed with a pseudohermitian structure. On the way we obtain a spectral interpretation of the Einstein-Hilbert action in pseudohermitian geometry.     

Algebras of symbols associated with the Weyl calculus for Lie group representations

We develop our earlier approach to the Weyl calculus for representations of infinite-dimensional Lie groups by establishing continuity properties of the Moyal product for symbols belonging to various modulation spaces. For instance, we prove that the modulation space of symbols M ^∞,1 is an associative Banach algebra and the corresponding operators are bounded. We then apply the abstract results to two classes of representations, namely the unitary irreducible representations of nilpotent Lie groups, and the natural representations of the semidirect product groups that govern the magnetic Weyl calculus. The classical Weyl–Hörmander calculus is obtained for the Schrödinger representations of the finite-dimensional Heisenberg groups, and in this case we recover the results obtained by J. Sjöstrand (Math Res Lett 1(2):185–192, 1994).     

One class of C*-algebras generated by a family of partial isometries and multiplicators

We consider a C*-subalgebra of the algebra of all bounded operators on the Hilbert space of square-summable functions defined on some countable set. The algebra under consideration is generated by a family of partial isometries and the multiplier algebra isomorphic to the algebra of all bounded functions defined on the mentioned set. The partial isometry operators satisfy correlations defined by a prescribed map on the set. We show that the considered algebra is ?-graduated. After that we construct the conditional expectation from the latter onto the subalgebra responding to zero. Using this conditional expectation, we prove that the algebra under consideration is nuclear.     

<Emphasis Type="Italic">p</Emphasis>-Adic Mikusinski calculus and its applications to the fourier and the mahler expansions of locally constant functions

We consider the p-adic counterpart of Mikusinski’s operational calculus based on the algebra C(ℤ _p ) of continuous functions on ℤ _p taking values in ℂ _p and equipped with the discrete Laplace convolution. Elements of the field (hyperfunctions) corresponding to shift operators, difference operators, and the indefinite sum operator are considered. A notion of p-adic exponent is generalized. Applications to the Fourier and the Mahler expansions of the indicator function of a ball and the convolution of two indicator functions are provided. Two ways of applying the p-adic analog of Mikusinski’s operational calculus lead us to the Fourier expansion for the fractional part of a p-adic number.     

Calculus of ordered operators on a factor algebra

We consider an algebra of operators in a Banach scale and its factor algebra modulo some power of a small parameter. On the factor algebra we construct an induced calculus of functions of ordered operators.     

Generalized Hille-Phillips type functional calculus for multiparameter semigroups

For generators of n-parameter strongly continuous operator semigroups in a Banach space, we construct a Hille-Phillips type functional calculus, the symbol class of which consists of analytic functions from the image of the Laplace transform of the convolution algebra of temperate distributions supported by the positive cone ? ₊ ⁿ . The image of such a calculus is described with the help of the commutant of the semigroup of shifts along the cone. The differential properties of the calculus and some examples are presented.