The goal of the present paper is to introduce and study noncommutative Hardy spaces associated with the regular $Λ$-polyball, to develop a functional calculus on noncommutative Hardy spaces for the completely non-coisometric (c.n.c.) $k$-tuples in ${\bf B}_Λ(H)$, and to study the characteristic functions and the associated multi-analytic models for the c.n.c. elements in the regular $Λ$-polyball. In addition, we show that the characteristic function is a complete unitary invariant for the class of c.n.c. $k$-tuples in ${\bf B}_Λ(H)$. These results extend the corresponding classical results of Sz.-Nagy--Foia\c s for contractions and the noncommutative versions for row contractions. In the particular case when $n_1=\cdots=n_k=1$ and $Λ_{ij}=1$, we obtain a functional calculus and operator model theory in terms of characteristic functions for $k$-tuples of contractions satisfying Brehmer condition.
In this paper, we study free k-pluriharmonic functions on noncommutative regular polyballs. These regular polyballs have universal operator models consisting of left creation operators acting on tensor products of full Fock spaces. We introduce and determine the class of kmulti- Toeplitz operators acting on these tensor products and show that the bounded free k-pluriharmonic functions on regular polyballs are precisely the noncommutative Berezin transforms of k-multi-Toeplitz operators. The Dirichlet extension problem on regular polyballs is also solved. It is proved that a free k-pluriharmoni…
In this paper, we study free holomorphic functions on regular polyballs and provide analogues of several classical results from complex analysis such as: Abel theorem, Hadamard formula, Cauchy inequality, Schwarz lemma, and maximum principle. These results are used together with a class of noncommutative Berezin transforms to obtain a complete description of the group Aut(B_n) of all free holomorphic automorphisms of the polyball. The abstract polyball B_n has a universal model S consisting of left creation operators acting on tensor products of full Fock spaces. We determine: the group of aut…
Multivariable operator theory is used to provide Bohr inequalities for free holomorphic functions with operator coefficients on the regular polyball. In addition, we obtain analogues of Caratheodory, Fejer, and Egervary-Szazs inequalities for free holomorhic functions with operator coefficients and positive real parts on the polyball. These results are used to provide multivariable analogues of Landau's inequality and Bohr's inequality when the norm is replaced by the numerical radius of an operator.
In this paper we develop a theory of curvature (resp. multiplicity) invariant for tensor products of full Fock spaces and also for tensor products of symmetric Fock spaces. This is an attempt to find a more general framework for these invariants and extend some of the results obtained by Arveson for the symmetric Fock space, by the author and Kribs for the full Fock space, and by Fang for the Hardy space over the polydisc. To prove the existence of the curvature and its basic properties in these settings requires a new approach based on noncommutative Berezin transforms and multivariable opera…
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 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 $\mathbf{R}^{2d}$.
Denoting by $\mathbb{M}$ the complexification of the quaternionic algebra $\mathbb{H}$, we characterize the family of those $\mathbb{M}$-valued functions, defined on subsets of $\H$, whose values are actually quaternions, using an intrinsic approach. In particular, we show that the slice quaternionic regularity can be characterized via a Cauchy type transform acting on the space of analytic $\mathbb{M}$-valued stem functions.
The aim of the present work is to show how, using the differential calculus associated to Dirichlet forms, it is possible to construct Fredholm modules on post critically finite fractals by regular harmonic structures. The modules are d-summable, the summability exponent d coinciding with the spectral dimension of the generalized laplacian operator associated with the regular harmonic structures. The characteristic tools of the noncommutative infinitesimal calculus allow to define a d-energy functional which is shown to be a self-similar conformal invariant.
We establish pathwise continuity properties of solutions to a stochastic Volterra equation with an additive noise term given by a local martingale. The deterministic part is governed by an operator with an $H^\infty$-calculus and a scalar kernel. The proof relies on the dilation theorem for positive definite operator families on a Hilbert space.
Gramsch and Lay [10] gave spectral mapping theorems for the Dunford-Taylor calculus of a closed linear operator $T$, $$\widetildeσ_i(f(T)) = f(\widetildeσ_i(T)), $$ for several extended essential spectra $\widetildeσ_i$. In this work, we extend such theorems for the natural functional calculus introduced by Haase [12,13]. We use the model case of bisectorial operators. The proofs presented here are generic, and are valid for similar 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 $H^\infty$-calculus on weighted $L^p$-spaces for power weights which fall outside the classical class of $A_p$-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.
Regarding quaternions as normal matrices, we first characterize the $2\times 2$ matrix-valued functions, defined on subsets of quaternions, whose values are quaternions. Then we investigate the regularity of quaternionic-valued functions, defined by the analytic functional calculus. Constructions of analytic functional calculi for real linear operators, in particular for quaternionic linear ones, are finally discussed, using a Riesz-Dunford-Gelfand type kernel in one variable, or a Martinelly type kernel in two variables.
The usual Weyl calculus is intimately associated with the choice of the standard symplectic structure on $\mathbb{R}^{n}\oplus\mathbb{R}^{n}$. In this paper we will show that the replacement of this structure by an arbitrary symplectic structure leads to a pseudo-differential calculus of operators acting on functions or distributions defined, not on $\mathbb{R}^{n}$ but rather on $\mathbb{R}^{n}\oplus\mathbb{R}^{n}$. These operators are intertwined with the standard Weyl pseudo-differential operators using an infinite family of partial isometries of $L^{2}(\mathbb{R}^{n})\longrightarrow L^{2}(…
We develop a very general operator-valued functional calculus for operators with an $H^{\infty}-$calculus. We then apply this to the joint functional calculus of two commuting sectorial operators when one has an $H^{\infty}$calculus. Using this we prove theorem of Dore-Venni type on sums of commuting sectorial operators and apply our results to the problem of $L_p-$maximal regularity. Our main assumption is the R-boundedness of certain sets of operators, and therefore methods from the geometry of Banach spaces are essential here. In the final section we exploit the special Banach space structu…
In the language of $L^\infty$-modules proposed by Gigli, we introduce a first order calculus on a topological Lusin measure space $(M,\mathfrak{m})$ carrying a quasi-regular, strongly local Dirichlet form $\mathscr{E}$. Furthermore, we develop a second order calculus if $(M,\mathscr{E},\mathfrak{m})$ is tamed by a signed measure in the extended Kato class in the sense of Erbar, Rigoni, Sturm and Tamanini. This allows us to define e.g. Hessians, covariant and exterior derivatives, Ricci curvature, and second fundamental form.
We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler-Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobi's theorem on conjugate points and Noether's theorem. We close with an application to low regularity Riemannian geometry.
The present note contains a review of $p$-energies and Sobolev spaces on metric measure spaces that carry a strongly local regular Dirichlet form. These Sobolev spaces are then used to generalize some basic results from the calculus of variations, such as the existence of minimizers for convex functionals and certain constrained mimimization problems. This applies to a number of non-classical situations such as degenerate diffusions, superpositions of diffusions and diffusions on fractals or on products of fractals.
The main purpose of this work is the construction of an analytic functional calculus for Clifford operators, which are operators acting on certain modules over Clifford algebras. Unlike in some preceding works by other authors, we use a spectrum defined in the complex plane, and certain stem functions, analytic in neighborhoods of such a spectrum. The replacement of the slice regular functions, having values in a Clifford algebra, by analytic stem functions becomes possible because of an isomorphism induced by a Cauchy type transform, whose existence is proved in the first part of this work.
Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to behave sub-multiplicatively under zigzag products. These results yield a combinatorial construction of super-expanders, i.e., a sequence of 3-regular graphs that does not admit a coarse embedding into any uniformly convex normed space.
