Analytic functions whose range is dense in a ball

Denote by Δ(resp. $\text{Δ}$) the open (resp. closed) unit disc in C. Let E be a closed subset of the unit circle T and let F be a relatively closed subset of T ? E of Lesbesgue measure zero. The following result is proved. Given a complex Banach space X and a bounded continuous function f:F → X, there exists an extension f? of f, bounded and continuous on $\text{}\text{?}\text{gD ? E}$, analytic on Δ and satisfying sup$\text{{6}\text{f}\text{?}\text{(z)6:zε}\text{δ}\text{?E}$. This is applied to show that for any separable complex Banach space X there exists an analytic function from Δ to X whose range is contained and dense in the unit ball of X.     

Spectral measures and the Bade reflexivity theorem

Let $\text{B}$ be a strongly equicontinuous Boolean algebra of projections on the quasi-complete locally convex space X and assume that the space L(X) of continuous linear operators on X is sequentially complete for the strong operator topology. Methods of integration with respect to spectral measures are used to show that the closed algebra generated by $\text{B}$ in L(X) consists precisely of those continuous linear operators on X which leave invariant each closed $\text{B}$-invariant subspace of X.     

On operator-valued analytic functions with constant norm

Let X be a complex Banach space and $\text{D}$ a domain in the complex plane. Let f: $\text{D}$ → X be an analytic function such that ∥f(ζ)∥ is constant as ζ ? $\text{D}$. If X is the complex plane, then by the classical maximum modulus theorem f;(ζ) itself is constant on $\text{D}$. This is not the case in general. In the paper we study the norm-constant analytic functions whose values are bounded linear operators over an uniformly convex complex Banach space or, in particular, over a complex Hilbert space.     

Unbounded operators generated by a given spectral measure

Let T be a closed densely-defined operator on a Banach space X and let E(·) be a spectral measure whose range $\text{E}$ is a complete Boolean algebra of projections in X. Then T is of the form ∝f(λ) dE(λ) if and only if T commutes with $\text{E}$ and leaves invariant every invariant subspace of $\text{E}$.     

Toeplitz matrices and determinants with Fisher-Hartwig symbols

If A and B are self-adjoint operators, this paper shows that A and B have order isomorphic invariant subspace lattices if and only if there are Borel subsets E and F of σ(A) and σ(B), respectively, whose complements have spectral measure zero, and there is a bijective function φ: E → F such that (i) Δ is a Borel subset of E if and only if φ(Δ) is a Borel subset of F; (ii) a Borel subset Δ of E has A-spectral measure zero if and only if φ(Δ) has B-spectral measure zero; (iii) B is unitarily equivalent to φ(A). If A is any self-adjoint operator, there is an associated function κ_A : $\text{N}$ ∪ {∞} → ($\text{N}$ ∪ {0, ∞}) × {0,1} defined in this paper. If $\text{F}$ denotes the collection of all functions from $\text{N}$ ∪ {∞} into ($\text{N}$ ∪ {0,∞}) × {0,1}, then $\text{F}$ is a parameter space for the isomorphism classes of the invariant subspace lattices of self-adjoint operators. That is, two self-adjoint operators A and B have isomorphic invariant subspace lattices if and only if κ_A = κ_B. The paper ends with some comments on the corresponding problem for normal operators.     

Functional calculus and duality for closed operators

In this paper, we continue our spectral-theoretic study [8] of unbounded closed operators in the framework of the spectral decomposition property and decomposable operators. Given a closed operator T with nonempty resolvent set, let f → f(T) be the homomorphism of the functional calculus. We show that if T has the spectral decomposition property, then f(T) is decomposable. Conversely, if f is nonconstant on every component of its domain which intersects the spectrum of T, then f(T) decomposable implies that T has the spectral decomposition property. A spectral duality theorems follows as a corollary. Furthermore, we obtain an analytic-type property for the canonical embedding J of the underlying Banach space X into its second dual $\text{X}{}^7$.     

Increasing and decreasing operators on complete lattices

The (isotone) map f: X → X is an increasing (decreasing) operator on the poset X if f(x) ? f²(x) (f²(x) ? f(x), resp.) holds for each x ∈ X. Properties of increasing (decreasing) operators on complete lattices are studied and shown to extend and clarify those of closure (resp. anticlosure) operators. The notion of the decreasing closure, $\text{f}$, (the increasing anticlosure, $\text{f}$,) of the map f: X → X is introduced extending that of the transitive closure, $\text{f}\text{?}$, of f. $\text{f}\text{f}$, and $\text{f}$ are all shown to have the same set of fixed points. Our results enable us to solve some problems raised by H. Crapo. In particular, the order structure of H(X), the set of retraction operators on X is analyzed. For X a complete lattice H(X) is shown to be a complete lattice in the pointwise partial order. We conclude by claiming that it is the increasing-decreasing character of the identity maps which yields the peculiar properties of Galois connections. This is done by defining a u-v connection between the posets X and Y, where u: X → X (v: Y → Y) is an increasing (resp. decreasing) operator to be a pair f, g of maps f; X → Y, g: Y → X such that gf ? u, fg ? v. It is shown that the whole theory of Galois connections can be carried over to u-v connections.     

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.     

On the square root of a positive B(X,X1)-valued function

Let (Ω, $\text{B}$, μ) be a measure space, $\text{X}$ a separable Banach space, and $\text{X}$¹ the space of all bounded conjugate linear functionals on $\text{X}$. Let f be a weak¹ summable positive B($\text{X, X}$¹)-valued function defined on Ω. The existence of a separable Hilbert space $\text{K}$, a weakly measurable B($\text{X, K}$)-valued function Q satisfying the relation $\text{Q}{}^1\text{(ω)Q(ω) = f(ω)}$ is proved. This result is used to define the Hilbert space L_2,f of square integrable operator-valued functions with respect to f. It is shown that for B⁺($\text{X, X}$¹)-valued measures, the concepts of weak¹, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained.     

Some generic properties of α-nonexpansive mappings

Let X be a Banach space, C a bounded closed subset of X, A a convex closed subset of X, $\text{E}$ a complete metric space formed by all α-nonexpansive mappings fC → A and $\text{M}$ a complete metric space formed by α-nonexpansive differentiable mappings fC → X. The following assertions are proved in this paper: (1) Properness of I ? f is a generic property in $\text{E}$ (2)the subset of $\text{E}$ formed by all α-contractive mappings is of Baire first category in $\text{E}$; and (3) for every y?X, the functional equation x ? f(x) = y has generically a finite number of solutions for f in $\text{M}$. Some applications to the fixed point theory and calculation of the topological degree are given.     

A note on Banach spaces containing c0 or C∞

It is shown that every separable Banach space X containing a subspace isomorphic to c₀ has a subspace Y with basis such that $\text{X}\text{Y}\text{~}\text{c}{}_0\text{C}{}_{\mathrm{\infty }}$ and the latter space has a shrinking basis and an unconditional FDD. Moreover, it is shown that X ⊕ C_∞ has a basis if X has the bounded approximation property.     

A class of extension properties that are not simply generated

Let $\text{P}$ be a closed-hereditary topological property preserved by products. Call a space $\text{P}$-regular if it is homeomorphic to a subspace of a product of spaces with $\text{P}$. Suppose that each $\text{P}$-regular space possesses a $\text{P}$-regular compactification. It is well-known that each $\text{P}$-regular space X is densely embedded in a unique space γ_scPX with $\text{P}$ such that if f: X → Y is continuous and Y has $\text{P}$, then f extends continuously to γ_scPX. Call $\text{P}$-pseudocompact if γ_scPX is compact.Associated with $\text{P}$ is another topological property $\text{P}$^#, possessing all the properties hypothesized for $\text{P}$ above, defined as follows: a $\text{P}$-regular space X has $\text{P}$^# if each $\text{P}$-pseudocompact closed subspace of X is compact. It is known that the $\text{P}$-pseudocompact spaces coincide with the $\text{P}$^#-pseudocompact spaces, and that $\text{P}$^# is the largest closed-hereditary, productive property for which this is the case. In this paper we prove that if $\text{P}$ is not the property of being compact and $\text{P}$-regular, then $\text{P}$^# is not simply generated; in other words, there does not exist a space E such that the spaces with $\text{P}$^# are precisely those spaces homeomorphic to closed subspaces of powers of E.     

Weakly compact composition operators on vector-valued BMOA

Weak compactness of the analytic composition operator f?f○φ is studied on BMOA(X), the space of X-valued analytic functions of bounded mean oscillation, and its subspace VMOA(X), where X is a complex Banach space. It is shown that the composition operator is weakly compact on BMOA(X) if X is reflexive and the corresponding composition operator is compact on the scalar-valued BMOA. A concrete example is given which shows that BMOA(X) differs from the weak vector-valued BMOA for infinite dimensional Banach spaces X.     

L2 multipliers with power weights

Necessary and sufficient conditions are found for a multiplier operator to be bounded on L² of the line with weight |x|^2α. This paper is concerned primarily with the case $\text{α>}\text{1}\text{2}$. Multiplier operators are defined on these spaces by using the usual definition on a subspace that is shown to be dense in the space. The case $\text{α < ?}\text{1}\text{2}$ is treated by duality; $\text{|α| <}\text{1}\text{2}$ is briefly treated using a recent result on fractional integrals. The periodic case is also sketched.     

Representations of differential operators on a Lie group

In this paper we apply the theory of second-order partial differential operators with nonnegative characteristic form to representations of Lie groups. We are concerned with a continuous representation U of a Lie group G in a Banach space $\text{B}$. Let $\text{E}$ be the enveloping algebra of G, and let dU be the infinitesimal homomorphism of $\text{E}$ into operators with the Gårding vectors as a common invariant domain. We study elements in $\text{E}$ of the form

$\text{P=}\text{∑}\text{1}\text{r}\text{X}{}^2{}_{\mathrm{j}}\text{|X}{}_0$

with the X_j,'s in the Lie algebra $\text{G}$.If the elements X₀, X₁,…, X_r generate $\text{G}$ as a Lie algebra then we show that the space of C^∞-vectors for U is precisely equal to the C^∞-vectors for the closure $\text{dU(P)}\text{,}\text{of}\text{dU(P)}$. This result is applied to obtain estimates for differential operators.The operator $\text{dU(P)}$ is the infinitesimal generator of a strongly continuous semigroup of operators in $\text{B}$. If X₀ = 0 we show that this semigroup can be analytically continued to complex time ζ with Re ζ > 0. The generalized heat kernels of these semigroups are computed. A space of rapidly decreasing functions on G is introduced in order to treat the heat kernels.For unitary representations we show essential self-adjointness of all operators $\text{dU(Σ}{}_1{}^{\mathrm{r}}\text{X}{}_{\mathrm{j}}{}^2\text{+ (?1)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{X}{}_0$ with X₀ in the real linear span of the X_j's. An application to quantum field theory is given.Finally, the new characterization of the C^∞-vectors is applied to a construction of a counterexample to a conjecture on exponentiation of operator Lie algebras.Our results on semigroups of exponential growth, and on the space of C^∞ vectors for a group representation can be viewed as generalizations of various results due to Nelson-Stinespring [18], and Poulsen [19], who prove essential self-adjointness and a priori estimates, respectively, for the sum of the squares of elements in a basis for $\text{G}$ (the Laplace operator). The work of Hörmander [11] and Bony [3] on degenerate-elliptic (hypoelliptic) operators supplies the technical basis for this generalization. The important feature is that elliptic regularity is too crude a tool for controlling commutators. With the aid of the above-mentioned hypoellipticity results we are able to “control” the (finite dimensional) Lie algebra generated by a given set of differential operators.     

Ergodic limits of tensor product semigroups

Let e^tSande^?tT be (C₀)-semigroups on a Banach space X. Their tensor product $\text{L}$(t) is defined by $\text{L}$(t)A = e^tSAe^tT (A?B(X)) and has the generator Δ formally of the form ΔA = SA ? AT. Under the assumption that {$\text{L}$(t); t ? 0} is bounded, we investigate the Abel limit and the Cesàro limit of $\text{L}$(t)A at ∞. If $\text{gW}{}_{\mathrm{s}}\text{[Ω}{}_{\mathrm{u}}\text{]}$ denotes the set of operators A for which the Abel limit P_s(A) [resp. P_u(A)] exists in the strong [resp. uniform] operator topology, then

$\text{N(Δ)⊕}\text{R(Δ)}\text{= ω}{}_{\mathrm{u}}\text{? ω}{}_{\mathrm{s}}\text{? N(Δ) +}\text{R(Δ)}$

and the limit defines a projection P_s[P_u] from $\text{Ω}{}_{\mathrm{s}}$ [resp. $\text{Ω}{}_{\mathrm{u}}$] onto N(Δ) with N(Δ) with $\text{R(Δ)}\text{= N(P}{}_{\mathrm{u}}\text{) ? N(P}{}_{\mathrm{u}}\text{) ?}\text{R(Δ)}$. If, in addition, S and T are Hilbert space normal operators such that _gq(S) ∩ _gq(T) ≠ φ, then $\text{Ω}{}_{\mathrm{u}}$ contains all compact operators.     

Banach spaces whose duals are isomorphic to l1(Γ)

Banach spaces X whose duals are isomorphic or isometric to l₁(Γ) are characterized by certain classes of operators on X. It is proved that a separable, conjugate space isomorphic to a complemented subspace of an L₁(S, Σ, μ) space is isomorphic to l₁; a $\text{L}$₁ space contained in a separable, conjugate space is isomorphic to a subspace of l₁.     

On differential operators with Bohr-Neugebauer type property

Let B be a bounded linear operator of a Banach space X into itself. If the differential operator $\text{(}\text{d}\text{dt}\text{) ? B}$ has a property more general than Bohr-Neugebauer property for Bochner almost-periodic functions, then any Stepanov-bounded solution of the differential equation $\text{(}\text{d}\text{dt}\text{) u(t) ? Bu(t) = g(t)}$ is also almost-periodic, with g(t) being continuous and Stepanov almost-periodic.     

Invariant subspaces of analytic multiparticle Hamiltonians

The quantum mechanics of n particles interacting through analytic two-body interactions can be formulated as a problem of functional analysis on a Hilbert space $\text{G}$ consisting of analytic functions. On $\text{G}$, there is an Hamiltonian H with resolvent R(λ). These quantities are associated with families of operators H(?) and R(λ, ?) on $\text{L}$, the case ? = 0 corresponding to standard quantum mechanics. The spectrum of H(?) consists of possible isolated points, plus a number of half-lines starting at the thresholds of scattering channels and making an angle 2? with the real axis.Assuming that the two-body interactions are in the Schmidt class on the two-particle space $\text{G}$, this paper studies the resolvent R(λ, ?) in the case ? ≠ 0. It is shown that a well known Fredholm equation for R(λ, ?) can be solved by the Neumann series whenever ¦λ¦ is sufficiently large and λ is not on a singular half-line. Owing to this, R(λ, ?) can be integrated around the various half-lines to yield bounded idempotent operators P_p(?) (p = 1, 2,…) on $\text{L}$. The range of P_p(?) is an invariant subspace of H(?). As ? varies, the family of operators P_p(?) generates a bounded idempotent operator P_p on a space $\text{G}$. The range of this is an invariant subspace of H. The relevance of this result to the problem of asymptotic completeness is indicated.