Holomorphic Functional Calculus Research Articles

Holomorphic functional calculus and vector-valued Littlewood–Paley–Stein theory for semigroups

Holomorphic functional calculus and vector-valued Littlewood–Paley–Stein theory for semigroups

On finitely summable Fredholm modules from Smale spaces

We prove that all K K -homology classes of the stable (and unstable) Ruelle algebra of a Smale space have explicit Fredholm module representatives that are finitely summable on the same smooth subalgebra and with the same degree of summability. The smooth subalgebra is induced by a metric on the underlying Smale space groupoid and fine transversality relations between stable and unstable sets. The degree of summability is related to the fractal dimension of the Smale space. Further, the Fredholm modules are obtained by taking Kasparov products with a fundamental class of the Spanier-Whitehead K K -duality between the Ruelle algebras. Finally, we obtain general results on stability under holomorphic functional calculus and construct Lipschitz algebras on étale groupoids.

Oversmoothing Tikhonov regularization in Banach spaces * *The work of DC was financially supported by National Natural Science Foundation of China (Nos. 11701205 and 11871240). The work of BH was financially supported by German Research Foundation (DFG Grant HO 1454/12-1). The work of IY was financially supported by German Research Foundation (DFG Grants YO159/2-2 and YO159/4-1).

This paper develops a Tikhonov regularization theory for nonlinear ill-posed operator equations in Banach spaces. As the main challenge, we consider the so-called oversmoothing state in the sense that the Tikhonov penalization is not able to capture the true solution regularity and leads to the infinite penalty value in the solution. We establish a vast extension of the Hilbertian convergence theory through the use of invertible sectorial operators from the holomorphic functional calculus and the prominent theory of interpolation scales in Banach spaces. Applications of the proposed theory involving ℓ 1, Bessel potential spaces, and Besov spaces are discussed.

An Introduction to Hyperholomorphic Spectral Theories and Fractional Powers of Vector Operators

The aim of this paper is to give an overview of the spectral theories associated with the notions of holomorphicity in dimension greater than one. A first natural extension is the theory of several complex variables whose Cauchy formula is used to define the holomorphic functional calculus for n-tuples of operators (A_1,ldots ,A_n). A second way is to consider hyperholomorphic functions of quaternionic or paravector variables. In this case, by the Fueter-Sce-Qian mapping theorem, we have two different notions of hyperholomorphic functions that are called slice hyperholomorphic functions and monogenic functions. Slice hyperholomorphic functions generate the spectral theory based on the S-spectrum while monogenic functions induce the spectral theory based on the monogenic spectrum. There is also an interesting relation between the two hyperholomorphic spectral theories via the F-functional calculus. The two hyperholomorphic spectral theories have different and complementary applications. We finally discuss how to define the fractional Fourier’s law for nonhomogeneous materials using the spectral theory on the S-spectrum.

A Weyl pseudodifferential calculus associated with exponential weights on Rd

We construct a Weyl pseudodifferential calculus tailored to studying boundedness of operators on weighted Lp spaces over Rd with weights of the form exp(−ϕ(x)), for ϕ a C2 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 Lp is determined, and its properties are analyzed. This theory is used to calculate an upper bounded on the H∞ angle of relevant operators and deduces known optimal results in some cases. Finally, the symbol class is enriched and studied under an algebraic viewpoint.

BMO spaces associated to operators with generalised Poisson bounds on non-doubling manifolds with ends

Consider a non-doubling manifold with ends M=Rn♯Rm where Rn=Rn×Sm−n for m>n≥3. We say that an operator L has a generalised Poisson kernel if L generates a semigroup e−tL whose kernel pt(x,y) has an upper bound similar to the kernel of e−tΔ where Δ is the Laplace-Beltrami operator on M. An example for operators with generalised Gaussian bounds is the Schrödinger operator L=Δ+V where V is an arbitrary non-negative locally integrable potential. In this paper, our aim is to introduce the BMO space BMOL(M) associated to operators with generalised Poisson bounds which serves as an appropriate setting for certain singular integrals with rough kernels to be bounded from L∞(M) into this new BMOL(M). On our BMOL(M) spaces, we show that the John–Nirenberg inequality holds and we show an interpolation theorem for a holomorphic family of operators which interpolates between Lq(M) and BMOL(M). As an application, we show that the holomorphic functional calculus m(L) is bounded from L∞(M) into BMOL(M), and bounded on Lp(M) for 1<p<∞.

Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients

We introduce a condition on accretive matrix functions, called $p$-ellipticity, and discuss its applications to the $L^p$ theory of elliptic PDE with complex coefficients. Our examples are: (i) generalized convexity of power functions (Bellman functions), (ii) dimension-free bilinear embeddings, (iii) $L^p$-contractivity of semigroups and (iv) holomorphic functional calculus. Recent work by Dindos and Pipher (arXiv:1612.01568v3) established close ties between $p$-ellipticity and (v) regularity theory of elliptic PDE with complex coefficients. The $p$-ellipticity condition arises from studying uniform positivity of a quadratic form associated with the matrix in question on one hand, and the Hessian of a power function on the other. Our results regarding contractivity extend earlier theorems by Cialdea and Maz'ya.

Noncommutative holomorphic functional calculus, affine and projective spaces from $\operatorname {NC}$-complete algebras

Noncommutative holomorphic functional calculus, affine and projective spaces from $\operatorname {NC}$-complete algebras

Formal Functional Calculus for Weakly Locally Nilpotent Operators in Fréchet Spaces

We generalize the classical Riesz–Dunford holomorphic functional calculus with weakly locally nilpotent operators in a Frechet space to the case of the algebra of all formal power series. We obtain a counterpart of the spectral mapping theorem and formulas similar to the Taylor and Riesz–Dunford formulas. We show that the weak local nilpotency of an operator is necessary for constructing the formal functional calculus. Bibliography: 12 titles.

The heat equation with rough boundary conditions and holomorphic functional calculus

In this paper we consider the Laplace operator with Dirichlet boundary conditions on a smooth domain. We prove that it has a bounded H∞-calculus on weighted Lp-spaces for power weights which fall outside the classical class of Ap-weights. Furthermore, we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity. In particular, we obtain a new maximal regularity result for the heat equation with rough inhomogeneous boundary data.

Bounded holomorphic functional calculus for nonsymmetric Ornstein-Uhlenbeck operators

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 $L^{2}(\gamma_{\infty})$, then ${\mathscr L}$ has bounded holomorphic functional calculus on $L^{r}(\gamma_{\infty})$, $1 \theta^{*}_{r}$, where $\gamma_{\infty}$ is the associated invariant measure and $\theta^{*}_{r}$ the sectoriality angle of ${\mathscr L}$ on $L^{r}(\gamma_{\infty})$. The angle $\theta^{*}_{r}$ is optimal. In particular our result applies to any nondegenerate finite dimensional Ornstein-Uhlenbeck operator, with dimension-free estimates.

Optimal angle of the holomorphic functional calculus for the Ornstein–Uhlenbeck operator

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 Lp(Rd,μ) for 1<p<∞, where μ is the standard Gaussian measure with density dμ=(2π)−d2exp(−|x|2∕2)dx. 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∞ functional calculus with this optimal angle.

Simultaneous Upper Triangular Forms for Commuting Operators in a Finite von Neumann Algebra

AbstractThe joint Brown measure and joint Haagerup–Schultz projections for tuples of commuting operators in a von Neumann algebra equipped with a faithful tracial state are investigated, and several natural properties are proved for these. It is shown that the support of the joint Brown measure is contained in the Taylor joint spectrum of the tuple, and also in the ostensibly smaller left Harte spectrum. A simultaneous upper triangularization result for finite commuting tuples is proved, and the joint Brown measure and joint Haagerup–Schultz projections are shown to behave well under the Arens multivariate holomorphic functional calculus of such a commuting tuple.

Frame decomposition and radial maximal semigroup characterization of Hardy spaces associated to operators

Let L be the generator of an analytic semigroup whose kernels satisfy Gaussian upper bounds and Hölder’s continuity. Also assume that L has a bounded holomorphic functional calculus on L2(Rn). In this paper, we construct a frame decomposition for the functions belonging to the Hardy space HL1(Rn) associated to L, and for functions in the Lebesgue spaces Lp, 1<p<∞. We then show that the corresponding HL1(Rn)-norm (resp. Lp(Rn)-norm) of a function f in terms of the frame coefficients is equivalent to the HL1(Rn)-norm (resp. Lp(Rn)-norm) of f. As an application of the frame decomposition, we establish the radial maximal semigroup characterization of the Hardy space HL1(Rn) under the extra condition of Gaussian upper bounds on the gradient of the heat kernels of L.

Hodge–Dirac, Hodge-Laplacian and Hodge–Stokes operators in $L^p$ spaces on Lipschitz domains

This paper concerns Hodge–Dirac operators $D\_{{}^\Vert}=d+\underline{\delta}$ acting in $L^p(\Omega, \Lambda)$ where $\Omega$ is a bounded open subset of ${\mathbb{R}}^n$ satisfying some kind of Lipschitz condition, $\Lambda$ is the exterior algebra of ${\mathbb{R}}^n$, $d$ is the exterior derivative acting on the de Rham complex of differential forms on $\Omega$, and $\underline{\delta}$ is the interior derivative with tangential boundary conditions. In $L^2(\Omega,\Lambda)$, $\underline{\delta} = {d}^\*$ and $D\_{{}^\Vert}$ is self-adjoint, thus having bounded resolvents ${({\rm I}+itD\_{{}^\Vert})^{-1}}{t\in{\mathbb{R}}}$ as well as a bounded functional calculus in $L^2(\Omega,\Lambda)$. We investigate the range of values $p\_H < p < p^H$ about $p=2$ for which $D{{}^\Vert}$ has bounded resolvents and a bounded holomorphic functional calculus in $L^p(\Omega,\Lambda)$. On domains which we call very weakly Lipschitz, we show that this is the same range of values as for which $L^p(\Omega,\Lambda)$ has a Hodge (or Helmholz) decomposition, being an open interval that includes 2. The Hodge-Laplacian $\Delta\_{{{}^\Vert}}$ is the square of the Hodge–Dirac operator, i.e., $-\Delta\_{{}^\Vert}={D\_{{}^\Vert}}^2$, so it also has a bounded functional calculus in $L^p(\Omega,\Lambda)$ when $p\_H < p < p^H$. But the Stokes operator with Hodge boundary conditions, which is the restriction of $-\Delta\_{{}^\Vert}$ to the subspace of divergence free vector fields in $L^p(\Omega,\Lambda^1)$ with tangential boundary conditions, has a bounded holomorphic functional calculus for further values of $p$, namely for max${1,{p\_H}\_S} < p < p^H$ where ${p\_H}\_S$ is the Sobolev exponent below $p\_H$, given by $1/{{p\_H}\_S} =1/{p\_H}+1/n$, so that ${{p\_H}\_S} < 2n/(n+2)$. In 3 dimensions, ${p\_H}\_S < 6/5$. We show also that for bounded strongly Lipschitz domains $\Omega$, $p\_H < 2n/(n+1) < 2n/(n-1) < p^H$, in agreement with the known results that $p\_H < 4/3 < 4 < p^H$ in dimension 2, and $p\_H < 3/2 < 3 < p^H$ in dimension 3. In both dimensions 2 and 3, ${p\_H}\_S<1$, implying that the Stokes operator has a bounded functional calculus in $L^p(\Omega,\Lambda^1)$ when $\Omega$ is strongly Lipschitz and $1 < p < p^H$.

Variable weak Hardy spaces WH L p(·)(ℝ n ) associated with operators satisfying Davies–Gaffney estimates

Abstract Let p ⁢ ( ⋅ ) : ℝ n → [ 0 , 1 ] {p(\,\cdot\,)\colon\mathbb{R}^{n}\to[0,1]} be a variable exponent function satisfying the globally log-Hölder continuous condition, and L a one-to-one operator of type ω in L 2 ⁢ ( ℝ n ) {L^{2}({\mathbb{R}}^{n})} , with ω ∈ [ 0 , π / 2 ) {\omega\in[0,\pi/2)} , which has a bounded holomorphic functional calculus and satisfies the Davies–Gaffney estimates. In this article, we introduce the variable weak Hardy space WH L p ⁢ ( ⋅ ) ⁢ ( ℝ n ) {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} , associated with L via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent space WT p ⁢ ( ⋅ ) ⁢ ( ℝ + n + 1 ) {\mathrm{WT}^{p(\,\cdot\,)}(\mathbb{R}_{+}^{n+1})} , which is also obtained in this article. In particular, when L is non-negative and self-adjoint, we obtain the atomic characterization of WH L p ⁢ ( ⋅ ) ⁢ ( ℝ n ) {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} . As an application of the molecular characterization, when L is the second-order divergence form elliptic operator with complex bounded measurable coefficients, we prove that the associated Riesz transform ∇ ⁡ L - 1 / 2 {\nabla L^{-1/2}} is bounded from WH L p ⁢ ( ⋅ ) ⁢ ( ℝ n ) {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} to the variable weak Hardy space WH p ⁢ ( ⋅ ) ⁢ ( ℝ n ) {\mathrm{WH}^{p(\,\cdot\,)}(\mathbb{R}^{n})} . Moreover, when L is non-negative and self-adjoint with the kernels of { e - t ⁢ L } t &gt; 0 {\{e^{-tL}\}_{t&gt;0}} satisfying the Gaussian upper bound estimates, the atomic characterization of WH L p ⁢ ( ⋅ ) ⁢ ( ℝ n ) {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} is further used to characterize this space via non-tangential maximal functions.

On continuity of solutions for parabolic control systems and input-to-state stability

We study minimal conditions under which mild solutions of linear evolutionary control systems are continuous for arbitrary bounded input functions. This question naturally appears when working with boundary controlled, linear partial differential equations. Here, we focus on parabolic equations which allow for operator-theoretic methods such as the holomorphic functional calculus. Moreover, we investigate stronger conditions than continuity leading to input-to-state stability with respect to Orlicz spaces. This also implies that the notions of input-to-state stability and integral-input-to-state stability coincide if additionally the uncontrolled equation is dissipative and the input space is finite-dimensional.

Analyticity of the Dirichlet-to-Neumann semigroup on continuous functions

Let $\Omega$ be a bounded open subset with $C^{1+\kappa}$-boundary for some $\kappa > 0$. Consider the Dirichlet-to-Neumann operator associated to the elliptic operator $- \sum \partial_l ( c_{kl} \, \partial_k ) + V$, where the $c_{kl} = c_{lk}$ are H\"older continuous and $V \in L_\infty(\Omega)$ are real valued. We prove that the Dirichlet-to-Neumann operator generates a $C_0$-semigroup on the space $C(\partial \Omega)$ which is in addition holomorphic with angle $\frac{\pi}{2}$. We also show that the kernel of the semigroup has Poisson bounds on the complex right half-plane. As a consequence we obtain an optimal holomorphic functional calculus and maximal regularity on $L_p(\Gamma)$ for all $p \in (1,\infty)$.

Beyond the spectral theorem: Spectrally decomposing arbitrary functions of nondiagonalizable operators

Nonlinearities in finite dimensions can be linearized by projecting them into infinite dimensions. Unfortunately, the familiar linear operator techniques that one would then hope to use often fail since the operators cannot be diagonalized. The curse of nondiagonalizability also plays an important role even in finite-dimensional linear operators, leading to analytical impediments that occur across many scientific domains. We show how to circumvent it via two tracks. First, using the well-known holomorphic functional calculus, we develop new practical results about spectral projection operators and the relationship between left and right generalized eigenvectors. Second, we generalize the holomorphic calculus to a meromorphic functional calculus that can decompose arbitrary functions of nondiagonalizable linear operators in terms of their eigenvalues and projection operators. This simultaneously simplifies and generalizes functional calculus so that it is readily applicable to analyzing complex physical systems. Together, these results extend the spectral theorem of normal operators to a much wider class, including circumstances in which poles and zeros of the function coincide with the operator spectrum. By allowing the direct manipulation of individual eigenspaces of nonnormal and nondiagonalizable operators, the new theory avoids spurious divergences. As such, it yields novel insights and closed-form expressions across several areas of physics in which nondiagonalizable dynamics arise, including memoryful stochastic processes, open nonunitary quantum systems, and far-from-equilibrium thermodynamics. The technical contributions include the first full treatment of arbitrary powers of an operator, highlighting the special role of the zero eigenvalue. Furthermore, we show that the Drazin inverse, previously only defined axiomatically, can be derived as the negative-one power of singular operators within the meromorphic functional calculus and we give a new general method to construct it. We provide new formulae for constructing spectral projection operators and delineate the relations among projection operators, eigenvectors, and left and right generalized eigenvectors. By way of illustrating its application, we explore several, rather distinct examples. First, we analyze stochastic transition operators in discrete and continuous time. Second, we show that nondiagonalizability can be a robust feature of a stochastic process, induced even by simple counting. As a result, we directly derive distributions of the time-dependent Poisson process and point out that nondiagonalizability is intrinsic to it and the broad class of hidden semi-Markov processes. Third, we show that the Drazin inverse arises naturally in stochastic thermodynamics and that applying the meromorphic functional calculus provides closed-form solutions for the dynamics of key thermodynamic observables. Finally, we draw connections to the Ruelle–Frobenius–Perron and Koopman operators for chaotic dynamical systems and propose how to extract eigenvalues from a time-series.

Variable Hardy Spaces Associated with Operators Satisfying Davies–Gaffney Estimates

AbstractLetLbe a one-to-one operator of type ω inL2(ℝn), with ω∈[0, π/2), which has a bounded holomorphic functional calculus and satisfies the Davies–Gaffney estimates. Letp(·): ℝn→(0, 1] be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors introduce the variable Hardy space$H_L^{p(\cdot )} ({\open R}^n)$associated withL. By means of variable tent spaces, the authors establish the molecular characterization of$H_L^{p(\cdot )} ({\open R}^n)$. Then the authors show that the dual space of$H_L^{p(\cdot )} ({\open R}^n)$is the bounded mean oscillation (BMO)-type space${\rm BM}{\rm O}_{p(\cdot ),{\kern 1pt} L^ * }({\open R}^n)$, whereL* denotes the adjoint operator ofL. In particular, whenLis the second-order divergence form elliptic operator with complex bounded measurable coefficients, the authors obtain the non-tangential maximal function characterization of$H_L^{p(\cdot )} ({\open R}^n)$and show that the fractional integralL−αfor α∈(0, (1/2)] is bounded from$H_L^{p(\cdot )} ({\open R}^n)$to$H_L^{q(\cdot )} ({\open R}^n)$with (1/p(·))−(1/q(·))=2α/n, and the Riesz transform ∇L−1/2is bounded from$H_L^{p(\cdot )} ({\open R}^n)$to the variable Hardy spaceHp(·)(ℝn).