一般多项式有界算子的泛函表示定理

该文在圆盘代数A(D)中引入了一个数乘变换, 找到了多项式有界算子的多项式演算与Riesz函数演算之间的联系, 得到了Banach空间X上的一般多项式有界算子的泛函表示定理.     

Feynman’s operational calculi for noncommuting operators: The monogenic calculus

In recent papers the authors presented their approach to Feynman’s operational calculi for a system of not necessarily commuting bounded linear operators acting on a Banach space. The central objects of the theory are the disentangling algebra, a commutative Banach algebra, and the disentangling map which carries this commutative structure into the noncommutative algebra of operators. Under assumptions concerning the growth of disentangled exponential expressions, the associated functional calculus for the system of operators is a distribution with compact support which we view as the joint spectrum of the operators with respect to the disentangling map. In this paper, the functional calculus is represented in terms of a higher-dimensional analogue of the Riesz-Dunford calculus using Clifford analysis.     

On some notions of convergence for n‐tuples of operators

The aim of this paper is to show that we can extend the notion of convergence in the norm‐resolvent sense to the case of several unbounded noncommuting operators (and to quaternionic operators as a particular case) using the notion of S‐resolvent operator. With this notion, we can define bounded functions of unbounded operators using the S‐functional calculus for n‐tuples of noncommuting operators. The same notion can be extended to the case of the F‐resolvent operator, which is the basis of the F‐functional calculus, a monogenic functional calculus for n‐tuples of commuting operators. We also prove some properties of the F‐functional calculus, which are of independent interest. Copyright © 2013 John Wiley & Sons, Ltd.     

The Cauchy formula with s-monogenic kernel and a functional calculus for noncommuting operators

The new notion of slice monogenic functions introduced in the paper [F. Colombo, I. Sabadini, D.C. Struppa, Slice monogenic functions, Israel J. Math. 171 (2009) 385-403] led us to define a new functional calculus for an n-tuple of not necessarily commuting operators, see [F. Colombo, I. Sabadini, D.C. Struppa, A new functional calculus for noncommuting operators, J. Funct. Anal. 254 (2008) 2255-2274]. In this paper we prove a Cauchy formula with slice monogenic kernel for the slice monogenic functions. This new Cauchy formula is the fundamental tool to prove that our functional calculus apply to a more general setting. Moreover, we deduce some fundamental properties of the functional calculus, for example: some algebraic properties, the Spectral Mapping Theorem and the Spectral Radius Theorem.     

Two Projection Theorems and Symbol Calculus for Operators with Massive Local Spectra

In this paper we construct a symbol calculus for Banach algebras generated by two idempotents and a coefficient algebra. This, combined with local principles for ?embedding algebras”?, leads to a symbol calculus for singular integral operators on spaces with Muckenhoupt weight and for singular integral operators with measurable coefficients.     

Bicomplex holomorphic functional calculus

In this paper we introduce and study a functional calculus for bicomplex linear bounded operators. The study is based on the decomposition of bicomplex numbers and of linear operators using the two nonreal idempotents. We show that, due to the presence of zero divisors in the bicomplex numbers, the spectrum of a bounded operator is unbounded. We therefore introduce a different spectrum (called reduced spectrum) which is bounded and turns out to be the right tool to construct the bicomplex holomorphic functional calculus. Finally we provide some properties of the calculus.     

The $H^{\infty}-$calculus and sums of closed operators

We develop a very general operator-valued functional calculus for operators with an calculus. We then apply this to the joint functional calculus of two sectorial operators when one has an calculus. Using this we prove theorem of Dore-Venni type on sums of sectorial operators and apply our results to the problem of 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 structure of spaces and spaces, to obtain some more detailed results in this setting. Received: 3 July 2000 / Revised version: 31 January 2001 / Published online: 23 July 2001     

Non Commutative Functional Calculus: Bounded Operators

In this paper we develop a functional calculus for bounded operators defined on quaternionic Banach spaces. This calculus is based on the new notion of slice-regularity, see Gentili and Struppa (Acad Sci Paris 342:741–744, 2006) and the key tools are a new resolvent operator and a new eigenvalue problem.     

A note on the functional calculus for unbounded self-adjoint operators

Two theorems of Riesz and Lorch (1936) are used to pass directly from the functional calculus for bounded symmetric operators to that for unbounded self-adjoint operators, thereby considerably shortening the passage via the spectral resolution for unbounded self-adjoint operators, and making particularly transparent the manner in which properties of the functional calculus for bounded operators are inherited by those which are unbounded.     

On Some Properties of the Quaternionic Functional Calculus

In some recent works we have developed a new functional calculus for bounded and unbounded quaternionic operators acting on a quaternionic Banach space. That functional calculus is based on the theory of slice regular functions and on a Cauchy formula which holds for particular domains where the admissible functions have power series expansions. In this paper, we use a new version of the Cauchy formula with slice regular kernel to extend the validity of the quaternionic functional calculus to functions defined on more general domains. Moreover, we show some of the algebraic properties of the quaternionic functional calculus such as the S-spectral radius theorem and the S-spectral mapping theorem. Our functional calculus is also a natural tool to define the semigroup e ^tA when A is a linear quaternionic operator.      

On some properties of the multidimensional Bochner-Phillips functional calculus

We develop the multidimensional functional calculus of semigroup generators which is based on the class of Bernstein functions in several variables. We establish spectral mapping theorems, give a holomorphy condition for the semigroups generated by the operators arising in this calculus, as well as prove the moment inequality for these operators.     

Distributed-order calculus: An operator-theoretic interpretation

Within the Bochner-Phillips functional calculus and Hirsch functional calculus, we describe the operators of distributed-order differentiation and integration as functions of the classical operators of differentiation and integration, respectively. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 4, pp. 478–486, April, 2008.     

The Fueter mapping theorem in integral form and the ℱ‐functional calculus

In this paper we show a version of the Fueter mapping theorem that can be stated in integral form based on the Cauchy formulas for slice monogenic (or slice regular) functions. More precisely, given a holomorphic function f of a paravector variable, we generate a monogenic function by an integral transform whose kernel is particularly simple. This procedure allows us to define a functional calculus for n‐tuples of commuting operators (called ?‐functional calculus) based on a new notion of spectrum, called ?‐spectrum, for the n‐tuples of operators. Analogous results are shown for the quaternionic version of the theory and for the related ?‐functional calculus. Copyright © 2010 John Wiley & Sons, Ltd.     

A Functional Calculus for n-Tuples of Noncommuting Operators

We employ the notion of slice monogenic functions to define a new functional calculus for an n-tuple of not necessarily commuting operators. This calculus is consistent with the Riesz-Dunford calculus for a single operator. Received: October, 2007. Accepted: February, 2008.     

A new functional calculus for noncommuting operators

In this paper we use the notion of slice monogenic functions [F. Colombo, I. Sabadini, D.C. Struppa, Slice monogenic functions, Israel J. Math., in press] to define a new functional calculus for an n-tuple T of not necessarily commuting operators. This calculus is different from the one discussed in [B. Jefferies, Spectral Properties of Noncommuting Operators, Lecture Notes in Math., vol. 1843, Springer-Verlag, Berlin, 2004] and it allows the explicit construction of the eigenvalue equation for the n-tuple T based on a new notion of spectrum for T. Our functional calculus is consistent with the Riesz-Dunford calculus in the case of a single operator.     

Functional Calculus and Joint Torsion of Pairs of Almost Commuting Operators

This paper investigates the transformation of determinants of pairs of Fredholm operators with trace class commutators. We study the extent to which the functional calculus commutes, modulo operator ideals, with projections in a finitely summable Fredholm module. As an application, we extend some results of Carey and Pincus on determinants of Toeplitz operators and Tate tame symbols. Additionally, we obtain variational formulas for joint torsion.     

Commutator Criteria for Magnetic Pseudodifferential Operators

The gauge covariant magnetic Weyl calculus has been introduced and studied in previous works. We prove criteria in terms of commutators for operators to be magnetic pseudodifferential operators of suitable symbol classes; neither the statements nor the proofs depend on a choice of a vector potential. We apply this criteria to inversion problems, functional calculus, affiliation results and to the study of the evolution group generated by a magnetic pseudodifferential operator.     

A note on the star order in Hilbert spaces

We study the star order on the algebra L(?) of bounded operators on a Hilbert space ?. We present a new interpretation of this order which allows to generalize to this setting many known results for matrices: functional calculus, semi-lattice properties, shorted operators and orthogonal decompositions. We also show several properties for general Hilbert spaces regarding the star order and its relationship with the functional calculus and the polar decomposition, which were unknown even in the finite-dimensional setting. We also study the existence of strong limits of star-monotone sequences and nets.     

Operators with smooth functional calculi

We introduce a class of (tuples of commuting) unbounded operators on a Banach space, admitting smooth functional calculi, which contains all operators of Helffer-Sjöstrand type and is closed under the action of smooth proper mappings. Moreover, the class is closed under tensor product of commuting operators. In general, and operator in this class has no resolvent in the usual sense, so the spectrum must be defined in terms of the functional calculus. We also consider invariant subspaces and spectral decompositions.     

Spectral Theorem for Definitizable Normal Linear Operators on Krein Spaces

In the present note a spectral theorem for normal definitizable linear operators on Krein spaces is derived by developing a functional calculus \({\phi \mapsto \phi(N)}\) which is the proper analogue of \({\phi \mapsto \int \phi \, dE}\) in the Hilbert space situation. This paper is the first systematical study of definitizable normal operators on Krein spaces.