Hilbert, Dirichlet and Fejér families of operators arising from C 0-groups, cosine functions and holomorphic semigroups

The main aim of this paper is to extend definitions of Hilbert transform, Dirichlet and Fejér operators (defined by convolution with suitable kernels in Lebesgue spaces) in arbitrary Banach spaces. We present a self-contained theory which includes different approaches of other authors whose starting points were usually C ₀-groups or cosine functions. We present relations with holomorphic semigroups. We characterize the geometric property of UMD spaces in terms of the Dirichlet and Fejér operators. To end the paper, we give examples to illustrate our results.     

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 group reduction for bounded cosine functions on UMD spaces

It is shown that if A generates a bounded cosine operator function on a UMD space X, then i(−A)^1/2 generates a bounded C ₀-group. The proof uses a transference principle for cosine functions.      

On typical properties of Hilbert space operators

We study the typical behavior of bounded linear operators on infinite-dimensional complex separable Hilbert spaces in the norm, strong-star, strong, weak polynomial and weak topologies. In particular, we investigate typical spectral properties, the problem of unitary equivalence of typical operators, and their embeddability into C ₀-semigroups. Our results provide information on the applicability of Baire category methods in the theory of Hilbert space operators.     

On Some Functions That Send Each Generator of a C0-Semigroup to the Generator of a Holomorphic Semigroup

We give some conditions on functions of the Schoenberg class T for them to send the generators of uniformly bounded semigroup of class C ₀ to the generators of holomorphic semigroups. This generalizes Yosida, Balakrishnan, and Kato's result relating to fractional powers of operators. The functional calculus of generators of C ₀-semigroups which uses the class T was constructed in the preceding articles of the author.     

Quasi-hyperbolic semigroups

We introduce the notion of quasi-hyperbolic operators and C₀-semigroups. Examples include the push-forward operator associated with a quasi-Anosov diffeomorphism or flow. A quasi-hyperbolic operator can be characterised by a simple spectral property or as the restriction of a hyperbolic operator to an invariant subspace. There is a corresponding spectral property for the generator of a C₀-semigroup, and it characterises quasi-hyperbolicity on Hilbert spaces but not on other Banach spaces. We exhibit some weaker properties which are implied by the spectral property.     

Poincaré and Opial inequalities for vector-valued convolution products

Various L^p$L^p$ form Poincaré and Opial inequalities are given for vector-valued convolution products. We apply our results to infinitesimal generators of _C0$C_0$-semigroups and cosine functions. Typical examples of these operators are differential operators in Lebesgue spaces.     

Closed embeddings of Hilbert spaces

Motivated by questions related to embeddings of homogeneous Sobolev spaces and to comparison of function spaces and operator ranges, we introduce the notion of closely embedded Hilbert spaces as an extension of that of continuous embedding of Hilbert spaces. We show that this notion is a special case of that of Hilbert spaces induced by unbounded positive selfadjoint operators that corresponds to kernel operators in the sense of L. Schwartz. Certain canonical representations and characterizations of uniqueness of closed embeddings are obtained. We exemplify these constructions by closed, but not continuous, embeddings of Hilbert spaces of holomorphic functions. An application to the closed embedding of a homogeneous Sobolev space on Rn in L₂(Rn), based on the singular integral operator associated to the Riesz potential, and a comparison to the case of the singular integral operator associated to the Bessel potential are also presented. As a second application we show that a closed embedding of two operator ranges corresponds to absolute continuity, in the sense of T. Ando, of the corresponding kernel operators.     

A New Approach to the Dore —Venni Theorem

The present paper works out the link between the Dore‐Venni theorem and the theory of analytic generators developped by I. Ciornescu and L. Zsid. The main result is an inverse theorem: on an UMD‐Banach space, analytic generators of C₀‐groups and operators with bounded imaginary powers are the same. The maximal regularity theorem of G. Dore and A. Venni appears as a corollary of this fact.     

Admissibility of Control Operators in UMD Spaces and the Inverse Laplace Transform

This paper investigates the admissibility of control and observation operators in UMD spaces. Necessary and/or sufficient conditions for unbounded control operators to be admissible and weakly admissible in the Salamon–Weiss sense are presented. This is illustrated by an example which shows that the UMD-property is essential. In particular, we get a direct proof of the known result of Driouich and and El-Mennaoui (Arch Math 72:56–63, 1999) on the validity of the inverse formula of the Laplace transform for C ₀-semigroups on UMD-spaces and in Hilbert spaces, as proved earlier by Yao (SIAM J Math Anal 26(5):1331–1341, 1995). We outline how these results can be used to prove a partial validity of the inverse Laplace transform for semigroups in general Banach spaces. In particular, we obtain the result on the inverse Laplace transform due to Hille and Philllips (Am Math Soc Transl Ser 2, 1957).     

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.     

A Landau-Kolmogorov inequality for generators of families of bounded operators

A Landau-Kolmogorov type inequality for generators of a wide class of strongly continuous families of bounded and linear operators defined on a Banach space is shown. Our approach allows us to recover (in a unified way) known results about uniformly bounded C₀-semigroups and cosine functions as well as to prove new results for other families of operators. In particular, if A is the generator of an α-times integrated family of bounded and linear operators arising from the well-posedness of fractional differential equations of order β+1 then, we prove that the inequality

     

The Umd Constants of the Summation Operators

The UMD property of a Banach space is one of the most useful properties when one thinks about possible applications. This is in particular due to the boundedness of the vector-valued Hilbert transform for functions with values in such a space. Looking at operators instead of at spaces, it is easy to check that the summation operator does not have the UMD property. The actual asymptotic behavior however of the UMD constants computed with martingales of length n is unknown. We explain, why it would be important to know this behavior, rephrase the problem of finding these UMD constants and give some evidence of how they behave asymptotically.     

A characteristic operator function for the class of n-hypercontractions

We consider a class of bounded linear operators on Hilbert space called n-hypercontractions which relates naturally to adjoint shift operators on certain vector-valued standard weighted Bergman spaces on the unit disc. In the context of n-hypercontractions in the class C0⋅ we introduce a counterpart to the so-called characteristic operator function for a contraction operator. This generalized characteristic operator function Wn,T is an operator-valued analytic function in the unit disc whose values are operators between two Hilbert spaces of defect type. Using an operator-valued function of the form Wn,T, we parametrize the wandering subspace for a general shift invariant subspace of the corresponding vector-valued standard weighted Bergman space. The operator-valued analytic function Wn,T is shown to act as a contractive multiplier from the Hardy space into the associated standard weighted Bergman space.     

On the relation of closed forms and Trotter's product formula

The aim of this paper is to give a characterization in Hilbert spaces of the generators of C₀-semigroups associated with closed, sectorial forms in terms of the convergence of a generalized Trotter's product formula. In the course of the proof of the main result we also present a similarity result which can be of independent interest: for any unbounded generator A of a C₀-semigroup e^tA it is possible to introduce an equivalent scalar product on the space, such that e^tA becomes non-quasi-contractive with respect to the new scalar product.     

A functional calculus for analytic generators ofC 0-groups

In [9] and [3] anF(S )-functional calculus for sectorial operators is constructed via the Dunford-Riesz integral. This calculus implicitely defines the well-known complex powers of such operators. Sectorial operators with bounded imaginary powers turn out to be of particular interest due to the remarkable Dore-Venni theorem. In [12] this theorem is proved via the theory of analytic generators ofC ₀-groups. These results suggest the existence ofF(S )-functional, calculi forC ₀-groups and their analytic generators. In this paper we show that such functional calculi indeed exsist, however the approach via the Dunford-Riesz integral is no longer viable. Therefore a different approach via an approximation argument is introduced. Existence and uniqueness theorems are given and we show how the functional calculi relate to known results. Examples illustrate the theory.     

Spectral properties of generators of theC 0-groups over tensor products of Banach spaces

We study the properties of common spectral subspaces of the N generators of one-parameter (C₀)-groups on different Banach spaces. The method of study is based on the functional calculus in the convolution algebra L¹ (R^N). We establish a theorem on the equality of the common spectra of the generators and their restrictions to common spectral subspaces.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 40–47.     

On interpolation of bilinear operators

In this paper we study interpolation of bilinear operators between products of Banach spaces generated by abstract methods of interpolation in the sense of Aronszajn and Gagliardo. A variant of bilinear interpolation theorem is proved for bilinear operators from corresponding weighted c₀ spaces into Banach spaces of non-trivial the periodic Fourier cotype. This result is then extended to the spaces generated by the well-known minimal and maximal methods of interpolation determined by quasi-concave functions. In the case when a maximal construction is generated by Hilbert spaces, we obtain a general variant of bilinear interpolation theorem. Combining this result with the abstract Grothendieck theorem of Pisier yields further results. The results are applied in deriving a bilinear interpolation theorem for Calderón-Lozanovsky, for Orlicz spaces and an embedding interpolation formula for (E,p)-summing operators.     

Kato's square root problem in Banach spaces

Let L be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces Lp(Rn;X) of X -valued functions on Rn. We characterize Kato's square root estimates and the H^∞-functional calculus of L in terms of R-boundedness properties of the resolvent of L, when X is a Banach function lattice with the UMD property, or a noncommutative Lp space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. In the special case X=C, we get a new approach to the Lp theory of square roots of elliptic operators, as well as an Lp version of Carleson's inequality.     

A perturbation theorem for operator semigroups in Hilbert spaces

We prove a perturbation result for C ₀-semigroups on Hilbert spaces and use it to show that certain operators of the form Au = iu ^(2k) + V · u ^(l) on L ² (?) generate a semigroup that is strongly continuous on (0, ∞).