Norms and spectral radii of composition operators acting on the Dirichlet space

We obtain new upper bounds on the norms of univalently induced composition operators acting on the Dirichlet space and compute explicitly the norms for univalent symbols whose range is the disk minus a set of measure zero. As an application, we show that the spectral radius of every univalently induced composition operator on the Dirichlet space is equal to one.     

On norms of composition operators acting on Bergman spaces

For arbitrary composition operators acting on a general Bergman space we improve the known lower bound for the norm and also generalize a related recent theorem of D.G. Pokorny and J.E. Shapiro. Next, we obtain a geometric formula for the norms of composition operators with linear fractional symbols, thus extending a result of C. Cowen and P. Hurst and revealing the meaning of their computation. Finally, we obtain a lower bound for essential norm of an arbitrary composition operator related to the well-known criterion of B. MacCluer and J.H. Shapiro. As a corollary, norms and essential norms are obtained for certain univalently induced noncompact composition operators in terms of the minimum of the angular derivative of the symbol.     

On Weighted Toeplitz, Big Hankel Operators and Carleson Measures

We compute the norm of pointwise multiplication operators, Toeplitz and Big Hankel operators with antiholomorphic symbols, defined on Besov spaces. These norms will be given in terms of Carleson measures for Besov spaces related to the symbol.     

Trace class norm inequalities for localization operators

We give sharp estimates on the norms in the trace class of localization operators in terms of their symbols.     

Weyl composition of symbols in large dimension

This paper is concerned with the Weyl composition of symbols in large dimension. We specify a class of symbols in order to estimate the Weyl symbol of the product of two Weyl h-pseudodifferential operators, with constants independent of the dimension. The proof includes regularized and hybrid compositions, together with a decomposition formula. We also analyze, in this context, the remainder term of the semiclassical expansion of the Weyl composition. The class of symbols contains symbols of Schrödinger semigroups in large dimension, typically for nearest neighbors or mean field interaction potentials. The Weyl composition is applied with Kac operators.     

ESSENTIAL NORMS OF PRODUCTS OF WEIGHTED COMPOSITION OPERATORS AND DIFFERENTIATION OPERATORS BETWEEN BANACH SPACES OF ANALYTIC FUNCTIONS

We obtain several estimates of the essential norms of the products of differentiation operators and weighted composition operators between weighted Banach spaces of analytic functions with general weights.As applications,we also give estimates of the essential norms of weighted composition operators between weighted Banach space of analytic functions and Bloch-type spaces.     

Weighted Hardy operators and commutators on Morrey spaces

The operator norms of weighted Hardy operators on Morrey spaces are worked out. The other purpose of this paper is to establish a sufficient and necessary condition on weight functions which ensures the boundedness of the commutators of weighted Hardy operators (with symbols in BMO(ℝ ⁿ )) on Morrey spaces.     

Non-vanishing functions and Toeplitz operators on tube-type domains

We prove an index theorem for Toeplitz operators on irreducible tube-type domains and we extend our results to Toeplitz operators with matrix symbols. In order to prove our index theorem, we proved a result asserting that a non-vanishing function on the Shilov boundary of a tube-type bounded symmetric domain, not necessarily irreducible, is equal to a unimodular function defined as the product of powers of generic norms times an exponential function.     

Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators

Bilinear operators are investigated in the context of Sobolev spaces and various techniques useful in the study of their boundedness properties are developed. In particular, several classes of symbols for bilinear operators beyond the so-called Coifman-Meyer class are considered. Some of the Sobolev space estimates obtained apply to both the bilinear Hilbert transform and its singular multipliers generalizations as well as to operators with variable dependent symbols. A symbolic calculus for the transposes of bilinear pseudodifferential operators and for the composition of linear and bilinear pseudodifferential operators is presented too.     

A note on weighted composition operators on the weighted Bergman space

This paper gives a note on weighted composition operators on the weighted Bergman space, which shows that for a fixed composition symbol, the weighted composition operators are bounded on the weighted Bergman space only with bounded weighted symbols if and only if the composition symbol is a finite Blaschke product.     

Subnormality and composition operators on the Bergman space

Following the work of C. Cowen and T. Kriete on the Hardy space, we prove that under a regularity condition, all composition operators with a subnormal adjoint on A²(D) have linear fractional symbols of the form. Moreover, we show that all composition operators on the Bergman space having these symbols have a subnormal adjoint, with larger range for the parameterr than found in the Hardy space case.     

Product of Volterra Type Integral and Composition Operators on Weighted Fock Spaces

We characterize the bounded, compact, and Schatten class product of Volterra type integral and composition operators acting between weighted Fock spaces. Our results are expressed in terms of certain Berezin type integral transforms on the complex plane ?. We also estimate the norms and essential norms of these operators in terms of the integral transforms. All our results are valid for weighted composition operators when acting between the class of weighted Fock spaces considered.     

Pseudo‐differential operators in a Gelfand–Shilov setting

We introduce some general classes of pseudodifferential operators with symbols admitting exponential type growth at infinity and we prove mapping properties for these operators on Gelfand–Shilov spaces. Moreover, we deduce composition and certain invariance properties of these classes.     

Symbolic Calculus and Fredholm Property for Localization Operators

We study the composition of time-frequency localization operators (wavepacket operators) and develop a symbolic calculus of such operators on modulation spaces. The use of time-frequency methods (phase space methods) allows the use of rough symbols of ultra-rapid growth in place of smooth symbols in the standard classes. As the main application it is shown that, in general, a localization operator possesses the Fredholm property, and thus its range is closed in the target space.     

Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions

The essential norm of weighted composition operators on weightedBanach spaces of analytic functions is computed in terms ofthe weights and the inducing symbols. As a consequence the boundednessand compactness of these operators is characterized. As anotherconsequence the essential norm of composition operators on weightedBloch spaces is obtained and, consequently, the boundednessand compactness of composition operators on these spaces isalso characterized. Particular instances of weighted Bloch spacesare the Lipschitz spaces. The method used allows a unified treatmentof the problem of boundedness and compactness on these spaces.     

Joint hyponormality of composition operators with linear fractional symbols

We study joint hyponormality and joint subnormality of ofn-tuples of commuting composition operators with linear fractional symbols, acting on the Hardy spaceH ². We also consider subnormality ofn-tuples of adjoints of composition operators.     

Pseudodifferential operators on<Emphasis Type="Italic">SU</Emphasis>(2)

We construct an algebra of left-invariant pseudodifferential operators on SU(2). We require only that the symbols be homogeneous and C². For Fourier-bandlimited symbols, we derive the expected formulae for composition and commutators and construct an orthonormal basis of common approximate eigenvectors that could be used to study spectral theory. Some remarks on applications to matrices of operators are made.     

Essential norms of weighted composition operators from weighted Bergman space to mixed-norm space on the unit ball

In this paper, we characterize the boundedness and compactness of the weighted composition operators from the weighted Bergman space to the standard mixed-norm space or the mixed-norm space with normal weight on the unit ball and estimate the essential norms of the weighted composition operators.     

Fourier Integral Operators Defined by Classical Symbols with Exit Behaviour

We present the detailed construction of the classical version of the calculus for Fourier Integral Operators (FIOs) in the class of symbols with exit behaviour ( SG symbols). In particular, we analyse what happens when one restricts the choice of amplitude and phase functions to the subset of the classical SG symbols. It turns out that the main composition theorem, obtained in the environment of general SG classes, has a “classical” counterpart. As an application, we study the Cauchy problem for classical hyperbolic operators of order (1, 1), refining the known results about the analogous problem for general SG hyperbolic operators. The theory developed here will be used in forthcoming papers to study the propagation of singularities and the Weyl formula for suitableclasses of operators defined on manifolds with cylindrical ends.