Hyponormality and subnormality for powers of commuting pairs of subnormal operators

Let H₀ (respectively H_∞) denote the class of commuting pairs of subnormal operators on Hilbert space (respectively subnormal pairs), and for an integer k?1 let Hk denote the class of k-hyponormal pairs in H₀. We study the hyponormality and subnormality of powers of pairs in Hk. We first show that if (T₁,T₂)∈H₁, the pair may fail to be in H₁. Conversely, we find a pair (T₁,T₂)∈H₀ such that but (T₁,T₂)∉H₁. Next, we show that there exists a pair (T₁,T₂)∈H₁ such that is subnormal (for all m,n?1), but (T₁,T₂) is not in H_∞; this further stretches the gap between the classes H₁ and H_∞. Finally, we prove that there exists a large class of 2-variable weighted shifts (T₁,T₂) (namely those pairs in H₀ whose cores are of tensor form (cf. Definition 3.4)), for which the subnormality of and does imply the subnormality of (T₁,T₂).     

On pure subnormal operators with finite rank self-commutators and related operator tuples

In this paper, we study the model of a pure subnormal operator with finite rank self-commutator and of the relatedn-tuple of commuting linear bounded operators. We also give some applications of the model to the theory ofn-tuples of commuting operators with trace class self-commutators.This work is supported in part by a NSF grant no. DMS-9400766.     

Joint spectrum of subnormal -tuples of composition operators

We compute the joint (Taylor) spectrum of an -tuple of commuting composition operators acting on the Hardy space

     

The product of generalized subnormal subgroups in finite groups

We study the structure of some classes of finite groups with a given system of commuting generalized subnormal subgroups. In particular, our results yield a characterization of superradical and Shemetkov formations.     

On pure subnormal semigroups

An example of a pure subbormal semigroup without commuting normal extension is given.Research supported by grants from the National Science Foundation. Portions of this work were done while the author was a Visiting Scientist at the Weizmann Institute of Science.     

On the commuting extensions of subnormal operators

We give some results concerning the following problem: Given a linear bounded operatorA which is subnormal on a Hilbert spaceH, andB its minimal normal extension on a Hilbert spaceK ～H, when can a quasi-normal operatorT commuting withA be extended to an operatorT ^e onK such thatT ^e commutes withB andT ^e is quasi-normal onK?     

Generalized spherical Aluthge transforms and binormality for commuting pairs of operators

In this paper, we introduce the notion of generalized spherical Aluthge transforms for commuting pairs of operators and study nontrivial joint invariant (resp. hyperinvariant) subspaces between the generalized spherical Aluthge transform and the original commuting pair. Next, we study the norm continuity through generalized Aluthge transform maps. We also study how the Taylor spectra and the Fredrolm index of commuting pairs of operators behave under the spherical Duggal transform. Finally, we introduce the notion of Campbell binormality for commuting pairs of operators and investigate some of its basic properties under spherical Aluthge and Duggal transforms. Moreover, we obtain new set inclusion diagrams among normal, quasinormal, centered, and Campbell binormal commuting pairs of operators.     

Joint hyponormality of composition operators with linear fractional symbols

We study joint hyponormality and joint subnormality of ofn-tuples of commuting composition operators with linear fractional symbols, acting on the Hardy spaceH ². We also consider subnormality ofn-tuples of adjoints of composition operators.     

Joint torsion of several commuting operators

We introduce the notion of joint torsion for several commuting operators satisfying a Fredholm condition. This new secondary invariant takes values in the group of invertibles of a field. It is constructed by comparing determinants associated with different filtrations of a Koszul complex. Our notion of joint torsion generalize the Carey–Pincus joint torsion of a pair of commuting Fredholm operators. As an example, under more restrictive invertibility assumptions, we show that the joint torsion recovers the multiplicative Lefschetz numbers. Furthermore, in the case of Toeplitz operators over the polydisc we provide a link between the joint torsion and the Cauchy integral formula. We will also consider the algebraic properties of the joint torsion. They include a cocycle property, a triviality property and a multiplicativity property. The proof of these results relies on a quite general comparison theorem for vertical and horizontal torsion isomorphisms associated with certain diagrams of chain complexes.     

On the representation of the joint spectrum for a commutingn-tuple of non-normal operators

The main purpose of the present paper is to describe the Taylor joint spectra forn-tuples of double commuting hyponormal operators, and to study the representation of the joint spectra in terms of that ofn-tuples of commuting normal operators.     

On the Fredholm and Weyl Spectra of Several Commuting Operators

In the paper the local structure of the Fredholm joint spectrum of commuting n-tuples of operators is considered. A connection between the spatial characteristics of operators and the algebraic invariants of the corresponding coherent sheaves is investigated. A new notion of Weyl joint spectrum of commuting n-tuple is introduced.     

Kernel identities for van Diejen’s q-difference operators and transformation formulas for multiple basic hypergeometric series

The kernel function of Cauchy type for type BC is defined as a solution of linear q-difference equations. In this paper, we show that this kernel function intertwines the commuting family of van Diejen’s q-difference operators. This result gives rise to a transformation formula for certain multiple basic hypergeometric series of type BC. We also construct a new infinite family of commuting q-difference operators for which the Koornwinder polynomials are joint eigenfunctions.     

Representation and character theory in 2-categories

We develop a (2-)categorical generalization of the theory of group representations and characters. We categorify the concept of the trace of a linear transformation, associating to any endofunctor of any small category a set called its categorical trace. In a linear situation, the categorical trace is a vector space and we associate to any two commuting self-equivalences a number called their joint trace. For a group acting on a linear category V we define an analog of the character which is the function on commuting pairs of group elements given by the joint traces of the corresponding functors. We call this function the 2-character of V. Such functions of commuting pairs (and more generally, n-tuples) appear in the work of Hopkins, Kuhn and Ravenel [Michael J. Hopkins, Nicholas J. Kuhn, Douglas C. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (3) (2000) 553-594 (electronic)] on equivariant Morava E-theories. We define the concept of induced categorical representation and show that the corresponding 2-character is given by the same formula as was obtained in [Michael J. Hopkins, Nicholas J. Kuhn, Douglas C. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (3) (2000) 553-594 (electronic)] for the transfer map in the second equivariant Morava E-theory.     

JOINT SPECTRAL SETS

Abstract

We consider the problem: if S,TEB(H) are commuting operators with von Neumann spectral sets X and Y respectively, does it imply that X x Y is a joint spectral set for the pair (S,T)?     

On the joint antieigenvalues of operators in normal subalgebras

We will study the joint antieigenvalues of pairs of operators that belong to the same closed normal subalgebra of B(H). This extends antieigenvalue theory from single normal operators to pairs of commuting normal operators.     

Spectral propeties of functions of commuting unbounded operators

We give generalizations of the joint spectral mapping theorem for certain classes of generally non-closed commuting operators on different complex Hilbert spaces. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 33–35.     

Expanding the joint spectrum of pairs of commuting contractions

We present a connection between solving the invariant subspace problem for a single operator on Hilbert space and the existence of a common invariant subspace for two commuting related operators. In particular, we reduce the problem of the existence of nontrivial invariant subspaces for a single contraction with spectral radius one to the problem of the existence of common nontrivial invariant subspaces for a pair of commuting contractions with large joint spectra.

     

Spectral theory for commutative algebras of differential operators on Lie groups

The joint spectral theory of a system of pairwise commuting self-adjoint left-invariant differential operators L₁,…,Ln on a connected Lie group G is studied, under the hypothesis that the algebra generated by them contains a “weighted subcoercive operator” of ter Elst and Robinson (1998) [52]. The joint spectrum of L₁,…,Ln in every unitary representation of G is characterized as the set of the eigenvalues corresponding to a particular class of (generalized) joint eigenfunctions of positive type of L₁,…,Ln. Connections with the theory of Gelfand pairs are established in the case L₁,…,Ln generate the algebra of K-invariant left-invariant differential operators on G for some compact subgroup K of Aut(G).     

On Taylor and other joint spectra for commuting n-tuples of operators

Let R and S be commuting n-tuples of operators. We will give some spectral relations between RS and SR that extend the case of single operators. We connect the Taylor spectrum, the Fredholm spectrum and some other joint spectra of RS and SR. Applications to Aluthge transforms of commuting n-tuples are also provided.     

Simultaneous triangularization of commuting matrices for the solution of polynomial equations

We present an extension of the QR method to simultaneously compute the joint eigenvalues of a finite family of commuting matrices. The problem is motivated by the task of finding solutions of a polynomial system. Several examples are included.