A criterion for an operator with real spectrum to be of scalar-type

Using a vector version of the Bochner—Schoenberg test which characterizes the Fourier—Stieltjes transforms of measures, a criterion is presented, in terms of the resolvent map, which determines when a linear operator with real spectrum acting in a locally convex space is a scalar-type spectral operator. This is an extension of a recent result of S. Kantorovitz.     

Well-bounded operators on nonreflexive Banach spaces

Every well-bounded operator on a reflexive Banach space is of type (B), and hence has a nice integral representation with respect to a spectral family of projections. A longstanding open question in the theory of well-bounded operators is whether there are any nonreflexive Banach spaces with this property. In this paper we extend the known results to show that on a very large class of nonreflexive spaces, one can always find a well-bounded operator which is not of type (B). We also prove that on any Banach space, compact well-bounded operators have a simple representation as a combination of disjoint projections.

     

On a calculus for a generalized scalar operator

Let $\text{X}$ be a Banach space; S and T bounded scalar-type operators in $\text{X}$. Define Δ on the space of bounded operators on $\text{X}$ by ΔX = TX ? XS if X is a bounded operator. We set up a calculus for Δ which allows us to consider f(Δ), for f a complex-valued bounded Borel measurable function on the spectrum of Δ, as an operator in the space of bounded operators whose domain is a subspace of operators which we call measure generating. This calculus is used to obtain some results on when the kernel of Δ is a complemented subspace of the space of bounded operators on $\text{X}$.     

Numerical Radius Norms on Operator Spaces

We introduce a numerical radius operator space (X, W_n). Theconditions to be a numerical radius operator space are weakerthan Ruan's axiom for an operator space (X, O_n). Let w(·)be the numerical radius on B(H). It is shown that, if X admitsa norm W_n(·) on the matrix space M_n(X) which satisfiesthe conditions, then there is a complete isometry, in the senseof the norms W_n(·) and w_n(·), from (X, W_n) into(B(H), w_n). We study the relationship between the operator space(X, O_n) and the numerical radius operator space (X, W_n). Thecategory of operator spaces can be regarded as a subcategoryof numerical radius operator spaces.     

Analytical Functional Models and Local Spectral Theory

In 1959 E. Bishop used a Banach-space version of the analyticduality principle established by e Silva, Köthe, Grothendieckand others to study connections between spectral decompositionproperties of a Banach-space operator and its adjoint. Accordingto Bishop a continuous linear operator T L(X) on a Banach spaceX satisfies property (rß) if the multiplication operator is injective with closed range for each open set U in the complex plane. In the present articlethe analytic duality principle in its original locally convexform is used to develop a complete duality theory for property(rß). At the same time it is shown that, up to similarity,property (rß) characterizes those operators occurringas restrictions of operators decomposable in the sense of C.Foias, and that its dual property, formulated as a spectraldecomposition property for the spectral subspaces of the givenoperator, characterizes those operators occurring as quotientsof decomposable operators. It is proved that, unlike the situationfor commuting subnormal operators, each finite commuting systemof operators with property (rß) can be extended toa finite commuting system of decomposable operators. Meanwhilethe results of this paper have been used to prove the existenceof invariant subspaces for subdecomposable operators with sufficientlyrich spectrum. 1991 Mathematics Subject Classification: 47A11,47B40.     

A Joint Functional Calculus for Sectorial Operators with Commuting Resolvents

In this paper we study the notion of joint functional calculusassociated with a couple of resolvent commuting sectorial operatorson a Banach space X. We present some positive results when Xis, for example, a Banach lattice or a quotient of subspacesof a B-convex Banach lattice. Furthermore, we develop a notionof a generalized H-functional calculus associated with the extensionto (H) of a sectorial operator on a B-convex Banach lattice, where H is a Hilbert space. We apply our results to a newconstruction of operators with a bounded H-functional calculusand to the maximal regularity problem. 1991 Mathematics SubjectClassification: 47A60, 47D06, 46C15.     

On scalar-type spectral operators and Carleman ultradifferentiable <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroups

Necessary and sufficient conditions for a scalar-type spectral operator in a Banach space to be a generator of a Carleman ultradifferentiable C ₀-semigroup are found. The conditions are formulated exclusively in terms of the spectrum of the operator. Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1215–1233, September, 2008.     

A Solution to the Invariant Subspace Problem

In this paper we exhibit a Banach space X, and a continuouslinear operator T: X X, such that X has no nontrivial closedT-invariant subspace.     

The Geometry of Convex Transitive Banach Spaces

Throughout this paper, X will denote a Banach space, S=S(X)and B=B(X) will be the unit sphere and the closed unit ballof X, respectively, and G=G(X) will stand for the group of allsurjective linear isometries on X. Unless explicitly statedotherwise, all Banach spaces will be assumed to be real. Nevertheless,by passing to real structures, the results remain true for complexspaces. 1991 Mathematics Subject Classification 46B04, 46B10,46B22.     

Operators on Subspaces of Hereditarily Indecomposable Banach Spaces

We show that if X is a complex hereditarily indecomposable space,then every operator from a subspace Y of X to X is of the formI + S, where I is the inclusion map and S is strictly singular.1991 Mathematics Subject Classification 46B20, 47B99.     

Logarithmic Terms in Trace Expansions of Atiyah–Patodi–Singer Problems

For Dirac-type operator D on a manifold X with a spectral boundarycondition (defined by a pseudodifferential projection), the associated heatoperator trace has an expansion in integer and half-integer powers and log-powersof t; the interest in the expansion coefficients goes back to the work of Atiyah,Patodi and Singer. In the product case considered by APS, it is known that allthe log-coefficients vanish when dim X is odd, whereas the log-coefficients atinteger powers vanish when dim X is even. We investigate here whether this partialvanishing of logarithms holds more generally. One type of result, shown forgeneral D with well-posed boundary conditions, is that a perturbation of Dby a tangential differential operator vanishing to order k on the boundaryleaves the first k log-power terms invariant (and the nonlocal power termsof the same degree are only locally perturbed). Another type of result is thatfor perturbations of the APS product case by tangential operators commuting withthe tangential part of D, all the logarithmic terms vanish when dim X is odd(whereas they can all be expected to be nonzero when dim X is even). The treatmentis based on earlier joint work with R. Seeley and a recent systematic parameter-dependentpseudodifferential boundary operator calculus, applied to the resolvent.     

Real Abelian Varieties with Many Line Bundles

Let X and Y be affine nonsingular real algebraic varieties.A general problem in real algebraic geometry is to try to decidewhen a continuous map f: X Y can be approximated by regularmaps in the space of c⁰ mappings from X to Y, equipped withthe c⁰ topology. This paper solves this problem when X is theconnected component containing the origin of the real part ofa complex Abelian variety and Y is the standard 2-dimensionalsphere.     

A classification theorem for essentially binormal operators

An essentially binormal operator on Hilbert space is an operator which is unitarily equivalent to a 2 × 2 matrix of essentially commuting, essentially normal operators. A natural invariant of essentially binormal operators up to unitary equivalence in the Calkin Algebra is the reducing essential 2 × 2 matricial spectrum. A nonempty compact subset X of the set of 2 × 2 matrices is called hypoconvex, if it is the reducing essential 2 × 2 matricial spectrum of an operator on Hilbert space. The set EN₂(X) is then defined to be the set of all equivalence classes (up to unitary equivalence in the Calkin algebra) of essentially binormal operators whose reducing essential 2 × 2 matricial spectrum coincides with X. The aim of this paper is to prove a result that enables one to compute EN₂(X) in terms of the topological structure of the space $\text{X}\text{?}$ of unitary orbits of X. Indeed, it is shown that for every hypoconvex subset X of the set of 2 × 2 matrices, there exists a natural homomorphism from $\text{Ext}\text{(}\text{X}\text{?}\text{)}$ onto EN₂(X). Also, a six term cyclic exact sequence is obtained, which produces a characterization of the kernel of the above-mentioned homomorphism.     

The Alternative Dunford-Pettis Property, Conjugations and Real Forms of C*-Algebras

Let be a conjugation, alias a conjugate linear isometry oforder 2, on a complex Banach space X and let X be the real formof X of -fixed points. In contrast to the Dunford–Pettisproperty, the alternative Dunford–Pettis property neednot lift from X to X. If X is a C^*-algebra it is shown thatX has the alternative Dunford–Pettis property if and onlyif X does and an analogous result is shown when X is the dualspace of a C^*-algebra. One consequence is that both Dunford–Pettisproperties coincide on all real forms of C^*-algebras.     

Solvability of symmetric word equations in positive definite letters

Let S(X, B) be a symmetric (‘palindromic’) wordin two letters X and B. A theorem due to Hillar and Johnsonstates that for each pair of positive definite matrices B andP, there is a positive definite solution X to the word equationS(X, B)=P. They also conjectured that these solutions are finiteand unique. In this paper, we resolve a modified version ofthis conjecture by showing that the Brouwer degree of such anequation is equal to 1 (in the case of real matrices). It followsthat, generically, the number of solutions is odd (and thusfinite) in the real case. Our approach allows us to addressthe more subtle question of uniqueness by exhibiting equationswith multiple real solutions, as well as providing a secondproof of the result of Hillar and Johnson in the real case.     

The Convex Hull of Extremal Vector-Valued Continuous Functions

Let T be a completely regular space and X a strictly convexn-dimensional real space. We prove that every continuous functionfrom T into the closed unit ball of X can be expressed as anaverage of eight continuous functions from T into the sphereof X if and only if dim (T) n–1, where dim(T) denotesthe covering dimension of T. The proof we give can be used toprove the same fact, without hypotheses on T, when X is infinite-dimensional,although in this case it has been proved recently that a betterresult can be obtained.     

Locally Accretive Mappings in Banach Spaces

Let X be a real Banach space for which the closed unit ballhas the fixed point property for nonexpansive self-mappings.Suppose that D is a bounded open subset of X, and T is a continuousmapping from the closure of D into X and locally accretive onD. Then T has a zero in D, provided that the following boundarycondition is fulfilled: there exists an element z in D so that||Tz|| < ||Tx|| for all x in the boundary of D.     

DEFORMATIONS OF HYPERCOMPLEX STRUCTURES ASSOCIATED TO HEISENBERG GROUPS

Let X be a compact quotient of the product of the real Heisenberggroup H_4m+1 of dimension 4m + 1 and the three-dimensional realEuclidean space R³. A left-invariant hypercomplex structureon H_4m+1 x R³ descends onto the compact quotient X. The spaceX is a hyperholomorphic fibration of 4-tori over a 4m-torus.We calculate the parameter space and obstructions to deformationsof this hypercomplex structure on X. Using our calculations,we show that all small deformations generate invariant hypercomplexstructures on X but not all of them arise from deformationsof the lattice. This is in contrast to the deformations on the4m-torus.     

POLYNOMIAL NUMERICAL INDEX FOR SOME COMPLEX VECTOR-VALUED FUNCTION SPACES

We study the relation between the polynomial numerical indicesof a complex vector-valued function space and the ones of itsrange space. It is proved that the spaces C(K, X) and L(µ,X) have the same polynomial numerical index as the complex Banachspace X for every compact Hausdorff space K and every -finitemeasure µ, which does not hold any more in the real case.We give an example of a complex Banach space X such that, forevery k 2, the polynomial numerical index of order k of X isthe greatest possible, namely 1, while the one of X** is theleast possible, namely k^k/(1–k). We also give new examplesof Banach spaces with the polynomial Daugavet property, namelyL(µ, X) when µ is atomless, and C_w(K, X), C_w*(K,X*) when K is perfect.