If X is a sequentially complete locally convex space, then a quotient bounded operator T is regular (in the sense of Waelbroeck) if and only if it is a bounded element (in the sense of Allan) of the algebra of quotient bounded operators on X. The classic functional calculus for the bounded operators on Banach space is generalized for such bounded operators. Also are studied some spectral propreties of the locally bounded operators.
We compare various functional calculus properties of Ritt operators. We show the existence of a Ritt operator T : X --> X on some Banach space X with the following property: T has a bounded $\H^\infty$ functional calculus with respect to the unit disc $\D$ (that is, T is polynomially bounded) but T does not have any bounded $\H^\infty$ functional calculus with respect to a Stolz domain of $\D$ with vertex at 1. Also we show that for an R-Ritt operator, the unconditional Ritt condition of Kalton-Portal is equivalent to the existence of a bounded $\H^\infty$ functional calculus with respect to s…
Spectral theory and functional calculus for unbounded self-adjoint operators on a Hilbert space are usually treated through von Neumann's Cayley transform. Based on ideas of Woronowicz, we redevelop this theory from the point of view of multiplier algebras and the so-called bounded transform (which establishes a bijective correspondence between closed operators and pure contractions). This also leads to a simple account of the affiliation relation between von Neumann algebras and self-adjoint operators.
In this article we apply a recently established transference principle in order to obtain the boundedness of certain functional calculi for semigroup generators. In particular, it is proved that if $-A$ generates a $C_0$-semigroup on a Hilbert space, then for each $τ>0$ the operator $A$ has a bounded calculus for the closed ideal of bounded holomorphic functions on a (sufficiently large) right half-plane that satisfy $f(z)=O(e^{-τ\textrm{Re}(z)})$ as $|z|\rightarrow \infty$. The bound of this calculus grows at most logarithmically as $τ\searrow 0$. As a consequence, $f(A)$ is a bounded operato…
In this article we study bounded operators $T$ on Banach space $X$ which satisfy the discrete Gomilko Shi-Feng condition $$\int_{0}^{2π}|\langle R(re^{it},T)^{2}x,x^*\rangle |dt \leq \frac{C}{(r^2-1)}\norme{x}\norme{x^*},\quad r>1, x\in X, x^* \in X^*. $$ We show that it is equivalent to a certain derivative bounded functional calculus and also to a bounded functional calculus relative to Besov space. Also on Hilbert space discrete Gomilko Shi-Feng condition is equivalent to power-boundedness. Finally we discuss the last equivalence on general Banach space involving the concept of $γ$-boundedn…
We study functional calculus properties of $C_{0}$-groups on real interpolation spaces, using transference principles. We obtain interpolation versions of the classical transference principle for bounded groups and of a recent transference principle for unbounded groups. Then we show that each group generator on a Banach space has a bounded $H^{\infty}_{1}$-calculus on real interpolation spaces. Additional results are derived from this.
We show that a bounded analytic semigroup on an $L_p$-space has a bounded $H^{\infty}(Σ_φ)$-calculus for some $φ< \fracπ{2}$ if and only if the semigroup can be obtained, after restricting to invariant subspaces, factorizing through invariant subspaces and similarity transforms, from a bounded analytic semigroup on some bigger $L_p$-space which is positive and contractive on the real line.
We observe that for a large class of non-amenable groups $G$, one can find bounded representations of $A(G)$ on Hilbert space which are not completely bounded. We also consider restriction algebras obtained from $A(G)$, equipped with the natural operator space structure, and ask whether such algebras can be completely isomorphic to operator algebras; partial results are obtained, using a modified notion of Helson set which takes account of operator space structure. In particular, we show that if $G$ is virtually abelian, then the restriction algebra $A_G(E)$ is completely isomorphic to an oper…
We study bounded holomorphic functional calculus for nonsymmetric infinite dimensional Ornstein-Uhlenbeck operators ${\mathscr L}$. We prove that if $-{\mathscr L}$ generates an analytic semigroup on $L^{2}(γ_{\infty})$, then ${\mathscr L}$ has bounded holomorphic functional calculus on $L^{r}(γ_{\infty})$, $1<r<\infty$, in any sector of angle $θ>θ^{*}_{r}$, where $γ_{\infty}$ is the associated invariant measure and $θ^{*}_{r}$ the sectoriality angle of ${\mathscr L}$ on $L^{r}(γ_{\infty})$. The angle $θ^{*}_{r}$ is optimal. In particular our result applies to any nondegenerate finite dimensio…
We prove that parameter-elliptic extensions of cone differential operators have a bounded $H_\infty$-calculus. Applications concern the Laplacian and the porous medium equation on manifolds with warped conical singularities.
We show that any bounded analytic semigroup on $L^p$ (with $1<p<\infty$) whose negative generator admits a bounded $H^{\infty}$ functional calculus with respect to some angle $< π/2$ can be dilated into a bounded analytic semigroup $(R_t)_{t\geq 0}$ on a bigger $L^p$-space in such a way that $R_t$ is a positive contraction for any $t$. We also establish a discrete analogue for Ritt operators and consider the case when $L^p$-spaces are replaced by more general Banach spaces. In connection with these functional calculus issues, we study isometric dilations of bounded continuous representations o…
We generalize the classical calculus rules satisfied by functions of bounded variation to the framework of RCD spaces. In the infinite dimensional setting we are able to define an analogue of the distributional differential and on finite dimensional spaces we prove fine properties and suitable calculus rules, such as the Vol'pert chain rule for vector valued functions.
We construct a calculus for generalized $\mathbf{SG}$ Fourier integral operators, extending known results to a broader class of symbols of $\mathbf{SG}$ type. In particular, we do not require that the phase functions are homogeneous. We also prove the $L^2(\mathbf{R}^{d})$-boundedness of the generalized $\mathbf{SG}$ Fourier integral operators having regular phase functions and amplitudes uniformly bounded on $\mathbf{R}^{2d}$.
In this article we give an approach to define continuous functional calculus for bounded quaternionic normal operators defined on a right quaternionic Hilbert space.
We develop a functional calculus for $d$-tuples of non-commuting elements in a Banach algebra. The functions we apply are free analytic functions, that is nc functions that are bounded on certain polynomial polyhedra.
We build on the work by Davies, extending the Helffer-Sjöstrand Functional Calculus domain for semi-bounded operators on Banach spaces given a priori controlled growth of the resolvents. We employ Seeley's Extension Theorem to extend smooth functions on the half line to the whole line and thus indirectly define functions of these operators.
In this note, we show that if a Banach space X has a predual, then every bounded linear operator on X with a continuous functional calculus admits a bounded Borel functional calculus. A consequence of this is that on such a Banach space, the classes of finitely spectral and prespectral operators coincide. We also apply this result to give some sufficient conditions for an operator with an absolutely continuous functional calculus to admit a bounded Borel one.
We investigate the boundedness of the $H^\infty$-calculus by estimating the bound $b(\varepsilon)$ of the mapping $H^{\infty}\rightarrow \mathcal{B}(X)$: $f\mapsto f(A)T(\varepsilon)$ for $\varepsilon$ near zero. Here, $-A$ generates the analytic semigroup $T$ and $H^{\infty}$ is the space of bounded analytic functions on a domain strictly containing the spectrum of $A$. We show that $b(\varepsilon)=\mathcal{O}(|\log\varepsilon|)$ in general, whereas $b(\varepsilon)=\mathcal{O}(1)$ for bounded calculi. This generalizes a result by Vitse and complements work by Haase and Rozendaal for non-analy…
We construct a new bounded functional calculus for the generators of bounded semigroups on Hilbert spaces and generators of bounded holomorphic semigroups on Banach spaces. The calculus is a natural (and strict) extension of the classical Hille-Phillips functional calculus, and it is compatible with the other well-known functional calculi. It satisfies the standard properties of functional calculi, provides a unified and direct approach to a number of norm-estimates in the literature, and allows improvements of some of them.
In this paper we present some spectral property for quotient bounded operators and locally bounded operators on locally convex spaces. We introduce the spectral radius of a quotient bounded operator and we show that the Gelfand formula for spectral radius and Newmann series have the same properties like in Banach spaces. Moreover, the geometrical radius of the spectrum is equal with the spectral radius. Using definitions of V.Troitsky we prove that the spectral set of a quotient bounded operator is compact set.
We construct a Weyl pseudodifferential calculus tailored to studying boundedness of operators on weighted $L^p$ spaces over $\mathbb{R}^d$ with weights of the form $\exp(-ϕ(x))$, for $ϕ$ a $C^2$ function, a setting in which the operator associated to the weighted Dirichlet form typically has only holomorphic functional calculus. A symbol class giving rise to bounded operators on $L^p$ is determined, and its properties analysed. This theory is used to calculate an upper bounded on the $H^\infty$ angle of relevant operators, and deduces known optimal results in some cases. Finally, the symbol cl…
In this paper, we first prove that the S-spectrum of a bounded right quaternionic linear operator on a two-sided quaternionic Banach space is a union of the spectrum of some bounded linear operators on a complex Banach space. Furthermore, we show that the S-functional calculus is obtained by the Riesz-Dunford functional calculus for complex linear operators. We also give simple proofs of some already existing results.
Given a Hilbert space and the generator $A$ of a strongly continuous, exponentially stable, semigroup on this Hilbert space. For any $g(-s) \in {\mathcal H}_{\infty}$ we show that there exists an infinite-time admissible output operator $g(A)$. If $g$ is rational, then this operator is bounded, and equals the "normal" definition of $g(A)$. In particular, when $g(s)=1/(s + α)$, $ α\in {\mathbb C}_0^+$, then this admissible output operator equals $(αI - A)^{-1}$. Although in general $g(A)$ may be unbounded, we always have that $g(A)$ multiplied by the semigroup is a bounded operator for every (s…
We derive conditions that ensure the existence of a bounded $H_\infty$-calculus in weighted $L_p$-Sobolev spaces for closed extensions $\underline{A}_T$ of a differential operator $A$ on a conic manifold with boundary, subject to differential boundary conditions $T$. In general, these conditions ask for a particular pseudodifferential structure of the resolvent $(λ-\underline{A}_T)^{-1}$ in a sector $Λ\subset\mathbf{C}$. In case of the minimal extension they reduce to parameter-ellipticity of the boundary value problem $(A,T)$. Examples concern the Dirichlet and Neumann Laplacians.
We study the boundedness of the $H^{\infty}$ functional calculus for differential operators acting in (L^{p}(\mathbb{R}^{n};\mathbb{C}^{N})). For constant coefficients, we give simple conditions on the symbols implying such boundedness. For non-constant coefficients, we extend our recent results for the (L^p) theory of the Kato square root problem to the more general framework of Hodge-Dirac operators with variable coefficients (Π_B) as treated in (L^2(\mathbb{R}^{n};\mathbb{C}^{N})) by Axelsson, Keith, and McIntosh. We obtain a characterization of the property that (Π_B) has a bounded (H^{\in…
We prove that the kernel of a quotient operator from an $\mathcal L_1$-space onto a Banach space $X$ with the Bounded Approximation Property (BAP) has the BAP. This completes earlier results of Lusky --case $\ell_1$-- and Figiel, Johnson and Pełczyński --case $X^*$ separable. Given a Banach space $X$, we show that if the kernel of a quotient map from some $\mathcal L_1$-space onto $X$ has the BAP then every kernel of every quotient map from any $\mathcal L_1$-space onto $X$ has the BAP. The dual result for $\mathcal L_\infty$-spaces also hold: if for some $\mathcal L_\infty$-space $E$ some quo…
This paper is devoted to the multivariable $H^\infty$ functional calculus associated with a finite commuting family of sectorial operators on Banach space. First we prove that if $(A_1,\ldots, A_d)$ is such a family, if $A_k$ is $R$-sectorial of $R$-type $ω_k\in(0,π)$, $k=1,\ldots,d$, and if $(A_1,\ldots, A_d)$ admits a bounded $H^\infty(Σ_{θ_1}\times \cdots\timesΣ_{θ_d})$ joint functional calculus for some $θ_k\in (ω_k,π)$, then it admits a bounded $H^\infty(Σ_{θ_1}\times \cdots\timesΣ_{θ_d})$ joint functional calculus for all $θ_k\in (ω_k,π)$, $k=1,\ldots,d$. Second we introduce square funct…
We establish a description of the maximal C*-algebra of quotients of a unital C*-algebra $A$ as a direct limit of spaces of completely bounded bimodule homomorphisms from certain operator submodules of the Haagerup tensor product $A\otimes_h A$ labelled by the essential closed right ideals of $A$ into $A$. In addition the invariance of the construction of the maximal C*-algebra of quotients under strong Morita equivalence is proved.
In this paper, we introduce a Weyl functional calculus $a \mapsto a(Q,P)$ for the position and momentum operators $Q$ and $P$ associated with the Ornstein-Uhlenbeck operator $ L = -Δ+ x\cdot \nabla$, and give a simple criterion for restricted $L^p$-$L^q$ boundedness of operators in this functional calculus. The analysis of this non-commutative functional calculus is simpler than the analysis of the functional calculus of $L$. It allows us to recover, unify, and extend, old and new results concerning the boundedness of $\exp(-zL)$ as an operator from $L^p(\mathbb{R}^d,γ_α)$ to $L^q(\mathbb{R}^d…
On a manifold $X$ with boundary and bounded geometry we consider a strongly elliptic second order operator $A$ together with a degenerate boundary operator $T$ of the form $T=φ_0γ_0 + φ_1γ_1$. Here $γ_0$ and $γ_1$ denote the evaluation of a function and its exterior normal derivative, respectively, at the boundary. We assume that $φ_0,φ_1\in C^{\infty}_b(\partial X)$, $φ_0,φ_1\ge 0$, and $φ_0+φ_1\geq c$, for some $c>0$. We also assume that the highest order coefficients of $A$ belong to $C^τ(X)$ for some $τ>0$ and the lower order coefficients are in $L_\infty(X)$. We show that the $L_p(X)$-rea…
For an operator generating a group on $L^p$ spaces transference results give bounds on the Phillips functional calculus also known as spectral multiplier estimates. In this paper we consider specific group generators which are abstraction of first order differential operators and prove similar spectral multiplier estimates assuming only that the group is bounded on $L^2$ rather than $L^p$. We also prove an R-bounded Hörmander calculus result by assuming an abstract Sobolev embedding property and show that the square of a perturbed Hodge-Dirac operator has such calculus.
We discuss the notion of $γ$-$H^{\infty}$-bounded calculus, strong $γ$-$m$-$H^{\infty}$-bounded calculus on half-plane and weak-$γ$-Gomilko-Shi-Feng condition and give a connection between them. Then we state a characterization of generation of $γ$-bounded $C_0$-semigroup in $K$-convex space, which leads to a version of Gearhart-Prüss on $K$-convex space.
We give a quantitative result about asymptotic moduli of Banach spaces under coarse quotient maps. More precisely, we prove that if a Banach space $Y$ is a coarse quotient of a subset of a Banach space $X$, where the coarse quotient map is coarse Lipschitz, then the ($β$)-modulus of $X$ is bounded by the modulus of asymptotic uniform smoothness of $Y$ up to some constants. In particular, if the coarse quotient map is a coarse homeomorphism, then the modulus of asymptotic uniform convexity of $X$ is bounded by the modulus of asymptotic uniform smoothness of $Y$ up to some constants.
By using an $H^{\infty}$ joint functional calculus for strongly commuting operators, we derive a scheme to deduce the $L^p$ boundedness of certain $d$-dimensional Riesz transforms from the $L^p$ boundedness of appropriate one-dimensional Riesz transforms. Moreover, the $L^p$ bounds we obtain are independent of the dimension. The scheme is applied to Riesz transforms connected with orthogonal expansions and discrete Riesz transforms on products of groups with polynomial growth.
In this article, we first study, in the framework of operator theory, Pusz and Woronowicz's functional calculus for pairs of bounded positive operators on Hilbert spaces associated with a homogeneous two-variable function on $[0,\infty)^2$. Our construction has special features that functions on $[0,\infty)^2$ are assumed only locally bounded from below and that the functional calculus is allowed to take extended semibounded self-adjoint operators. To analyze convexity properties of the functional calculus, we extend the notion of operator convexity for real functions to that for functions wit…
A Ritt operator T : X --> X on Banach space is a power bounded operator such that the sequence of all n(T^{n} -T^{n-1}) is bounded. When X=Lp for some 1<p<\infty, we study the validity of square functions estimates Norm{(\sum_k k |T^{k}(x) - T^{k-1}(x)|^2)^{1/2}}_{Lp} \leq K Norm{x}_{Lp} for such operators. We show that T and its adjoint T^* both satisfy such estimates if and only if T admits a bounded functional calculus with respect to a Stolz domain. This is a single operator analog of the famous Cowling-Doust-McIntosh-Yagi characterization of bounded H^\infty-calculus on $Lp$-spaces by the…
Beurling--Kato theory, Hardy--Sobolev calculus and Ritt operators
Aix-Marseille University（艾克斯-马赛大学）；CNRS, I2M Marseille（法国国家科学研究中心，I2M马赛）；Nicolaus Copernicus University（尼古拉·哥白尼大学）；Institute of Mathematics Polish Academy of Sciences（波兰科学院数学研究所）
Using notions from the geometry of Banach spaces we introduce square functions $γ(Ω,X)$ for functions with values in an arbitrary Banach space $X$. We show that they have very convenient function space properties comparable to the Bochner norm of $L_2(Ω,H)$ for a Hilbert space $H$. In particular all bounded operators $T$ on $H$ can be extended to $γ(Ω,X)$ for all Banach spaces $X$. Our main applications are characterizations of the $H^{\infty}$--calculus that extend known results for $L_p$--spaces from \cite{CowlingDoustMcIntoshYagi}. With these square function estimates we show, e. g., that a…
In this paper the notion of an abstract square function (estimate) is introduced as an operator X to gamma (H; Y), where X, Y are Banach spaces, H is a Hilbert space, and gamma(H; Y) is the space of gamma-radonifying operators. By the seminal work of Kalton and Weis, this definition is a coherent generalisation of the classical notion of square function appearing in the theory of singular integrals. Given an abstract functional calculus (E, F, Phi) on a Banach space X, where F (O) is an algebra of scalar-valued functions on a set O, we define a square function Phi_gamma(f) for certain H-valued…
Let T_t = e^{-tA} be a bounded analytic semigroup on Lp, with 1<p<\infty. It is known that if A and its adjoint A^* both satisfy square function estimates \bignorm{\bigl(\int_{0}^{\infty}| A^{1/2} T_t(x)|^2\, dt\,\bigr)^{1/2}_{Lp} \lesssim \norm{x} and \bignorm{\bigl(\int_{0}^{\infty}|A^{*}^{1/2} T_t^*(y)|^2\, dt\,\bigr)^{1/2}_{Lp'} \lesssim \norm{y} for x in Lp and y in Lp', then A admits a bounded H^{\infty}(Σ_θ) functional calculus for any θ>\fracπ{2}. We show that this actually holds true for some θ<\fracπ{2}.
This note deals with the boundedness of the $H^\infty$ functional calculus of Ritt operators $T$ and associated square function estimates. The purpose is to give a shorter, concise and slightly more general approach towards Le~Merdy's results [19] on this subject.
We prove a transference principle for general (i.e., not necessarily bounded) strongly continuous groups on Banach spaces. If the Banach space has the UMD property, the transference principle leads to estimates for the functional calculus of the group generator. In the Hilbert space case, the results cover classical theorems of McIntosh and Boyadzhiev-de Laubenfels; in the UMD case they are analogues of classical results by Hieber and Pruess. By using functional calculus methods, consequences for sectorial operators are derived. For instance it is proved, that every generator of a cosine funct…
Let ${\cal{M}}$ be a complete Riemannian manifold with Ricci curvature bounded below and Laplace operator $Δ$. The paper develops a functional calculus for the cosine family $\cos(t\sqrt Δ)$ which is associated with waves that travel at unit speed. If $f$ is holomorphic on a Venturi shaped region, and $z^kf(z)$ is bounded for some positive integer $k$, then $f({\sqrt Δ})$ defines a bounded linear operator on $L^p({\cal{M}})$ for some $p>2$. For Jacobi hypergroups with invariant measure $m$ the generalized Fourier transform of $f\in L^1(m)$ gives $\hat f\in H^\infty (Σ_ω)$ for some strip $Σ_ω$.…
We establish the existence of a bounded $H_\infty$-calculus for a large class of hypoelliptic pseudodifferential operators on R^n and closed manifolds.
We show that, given a Banach space and a generator of an exponentially stable $C_{0}$-semigroup, a weakly admissible operator $g(A)$ can be defined for any $g$ bounded, analytic function on the left half-plane. This yields an (unbounded) functional calculus. The construction uses a Toeplitz operator and is motivated by system theory. In separable Hilbert spaces, we even get admissibility. Furthermore, it is investigated when a bounded calculus can be guaranteed. For this we introduce the new notion of exact observability by direction. Finally, it is shown that the calculus coincides with one f…
We prove that the existence of a Mihlin-Hörmander functional calculus for an operator $L$ implies the boundedness on $L^p$ of both the maximal operators and the continuous square functions build on spectral multipliers of $L.$ The considered multiplier functions are finitely smooth and satisfy an integral condition at infinity. In particular multipliers of compact support are admitted.
The wark is a contribution to the functional calculus constructed by of the author and A. A. Atvinovskii of closed operators on Banach spaces. The calculus is based on Markov and related functions as symbols. Estimates of bounded perturbations of operator functions with respect to general operator ideal norms, operator Lipschitzness, moment inequality, Freshet operator differentiability, and Livshitz-Krein trace formulae have been considered.
We give a simple proof of the fact that the classical Ornstein-Uhlenbeck operator $L$ is R-sectorial of angle $arcsin|1-2/p|$ on $L^{p}(\mathbb{R}^{n},\exp(-|x|^2/2)dx)$ (for $1<p<\infty$). Applying the abstract holomorphic functional calculus theory of Kalton and Weis, this immediately gives a new proof of the fact that $L$ has a bounded $H^{\infty}$ functional calculus with this optimal angle.
In this paper we develop a new set of results based on a nonlocal gradient jointly inspired by the Riesz s-fractional gradient and Peridynamics, in the sense that its integration domain depends on a ball of radius delta > 0 (horizon of interaction among particles, in the terminology of Peridynamics), while keeping at the same time the singularity of the Riesz potential in its integration kernel. Accordingly, we define a functional space suitable for nonlocal models in Calculus of Variations and partial differential equations. Our motivation is to develop the proper functional analysis framewor…
In this paper, we study some features of n-normed spaces with respect to norms of its quotient spaces. We define continuous functions with respect to the norms of its quotient spaces and show that all types of continuity are equivalent. We also study contractive mappings on n- normed spaces using the same approach. In particular, we prove a fixed point theorem for contractive mappings on a closed and bounded set in an n-normed space.
We prove spectral multiplier theorems for Hörmander classes $\mathcal{H}^α\_p$ for 0-sectorial operators A on Banach spaces assuming a bounded $H^\infty(Σ\_σ)$ calculus for some $σ\in (0,π)$ and norm and certain R-bounds on one of the following families of operators: the semigroup $e^{--zA}$ on $\mathbb{C}\_+$, the wave operators $e^{isA}$ for $s \in \mathbb{R}$, the resolvent $(λ-- A)^{-1}$ on $\mathbb{C} \backslash \mathbb{R}$, the imaginary powers $A^{it}$ for $t \in \mathbb{R}$ or the Bochner-Riesz means $(1-A/u)^α\_+$ for $u > 0.$ In contrast to the existing literature we neither assume t…
Given any square matrix or a bounded operator $A$ in a Hilbert space such that $p(A)$ is normal (or similar to normal), we construct a Banach algebra, depending on the polynomial $p$, for which a simple functional calculus holds. When the polynomial is of degree $d$, then the algebra deals with continuous $\mathbb C^d$-valued functions, defined on the spectrum of $p(A)$. In particular, the calculus provides a natural approach to deal with nontrivial Jordan blocks and one does not need differentiability at such eigenvalues.
In this paper, we study some topological characteristics of the n-normed spaces. We observe convergence sequences, closed sets, and bounded sets in the n-normed spaces using norms of quotient spaces that will be constructed. These norms will be a new viewpoint in observing the characteristics of the n-normed spaces. By using these norms, we also review the completeness of the n-normed spaces.
