James Jamison

The University of Memphis | U of M · Department of Mathematical Sciences

In this paper we give a representation for the surjective and linear isometries on spaces of vector valued absolutely continuous functions with p-integrable derivatives (1 < p < ∞). The range space is a Banach space which is smooth, reflexive and separable. Representations for the generalized bi-circular projections and the bounded hermitian operat...

In this paper we investigate the algebraic structure of the isometry group of several classical Banach spaces, namely, symmetric sequence spaces, ( ), , , and bounded operators endowed with Chan-Li-Tui unitarily invariant -norms. We also identify the isometrically invariant subspaces for each of theses settings.

We study algebraic and topological properties of the group of all surjective isometrics on several spaces of vector valued analytic functions and vector valued L-p spaces (1 <= p <= infinity). We also derive the form for the surjective linear isometrics on the vector valued little Zygmund space.

For compact metric spaces (X,d), we show that the Lipschitz spaces Lip (X,d) and the little Lipschitz spaces lip (X,d α ) with 0<α<1, equipped with the sum norm, support only trivial hermitian operators, that is, real multiples of the identity operator.

In this paper we describe the surjective linear isometries on a vector valued little Bloch space with range space a strictly convex and smooth complex Banach space. We also describe the hermitian operators and the generalized bi-circular projections supported by these spaces.

We establish the algebraic reflexivity of three isometry groups of operator structures: the group of all surjective isometries on the unitary group, the group of all surjective isometries on the set of all positive invertible operators equipped with the Thompson metric, and the group of all surjective isometries on the general linear group of , the...

Let H be a complex Hilbert space, n a given positive integer and let Pn(H) be the set of all projections on H with rank n. Under the condition dimH≥4n, we describe the surjective isometries of Pn(H) with respect to the gap metric (the metric induced by the operator norm).

In this article the notion of circular operator is extended to the Banach space setting. In particular, this property is considered for elementary operators of lengths one and two acting on minimal norm ideals of ℬ(ℋ). Necessary and sufficient conditions for the circularity of generalized derivations and Lüders operators are also obtained.

For X a compact Hausdorff topological space and F a strictly convex and reflexive Banach space we characterize the surjective isometries of certain subspaces of C(X, F). It follows from this characterization that surjective isometries on spaces of vector valued Lipschitz functions equipped with a p-norm are also weighted composition operators.

We derive representations for homomorphisms and isomorphisms between Banach algebras of Lipschitz functions with values in a sequence space, including l. We show that such homomorphisms are automatically continuous and preserve the ? operation. We also give necessary conditions for the compactness of homomorphisms in these settings and give charact...

This paper gives a characterization of a class of surjective isometries on spaces of Lipschitz functions with values in a finite dimensional complex Hilbert space.

This paper characterizes the hermitian operators on spaces of Banach- valued Lipschitz functions.

We prove several results concerning the representation of projections on arbitrary Banach spaces. We also give illustrative examples including an example of a generalized bi-circular projection which can not be written as the average of the identity with an isometric reflection. We also characterize generalized bi-circular projections on $C_0(\Om,X...

We consider the elementary operator L , acting on the Hilbert–Schmidt class C 2 ( H ) , given by L ( T ) = ATB , with A and B bounded operators on a separable Hilbert space H . In this paper we establish results relating isometric properties of L with those of the defining symbols A and B . We also show that if A is a strict n -isometry on a Hilber...

We characterize projections on spaces of Lipschitz functions expressed as the average of two and three linear surjective isometries. Generalized bi-circular projections are the only projections on these spaces given as the convex combination of two surjective isometries.

In this paper we consider thickness and thinness of the unit sphere for infinite dimensional Banach spaces, as proposed by Whitley in [11]. We compute the thickness of several outer sums of Banach spaces and derive conclusions about the almost Daugavet property for those spaces. We also find estimates of the thickness of unit sphere in L p spaces,...

For a Banach space E and a compact metric space (X,d), a function F:X→E is a Lipschitz function if there exists k>0 such that‖F(x)−F(y)‖⩽kd(x,y)for all x,y∈X. The smallest such k is called the Lipschitz constant L(F) for F. The space Lip(X,E) of all Lipschitz functions from X to E is a Banach space under the norm defined by‖F‖=max{L(F),‖F‖∞}, where...

In this paper, we consider projections on minimal norm ideals of B(H) that are represented as the average of two surjective isometries. We describe projections of the formP(T)=A1TB1+A2TB22,where A1, A2, B1 and B2 are unitary operators satisfying a commutativity condition. We also characterize classes of projections of the formP(T)=AT+TtB2andP(T)=AT...

The Nakano space Lp(t)(μ) associated with p(t) is defined to be the Musielak-Orlicz space Lφφ(μ) such that φ (u,t) = u p(t)/p(t). We are going to consider the space N = L p(t)(μ,H), where H is a separable complex Hilbert space with inner product (,) and norm ∥·∥2 For any f ∈ N, let Mathmatical equation representative where 1 < po ≤ p(t) ≤ P∞ ≤ ∞. F...

We give a characterization of generalized bicircular projections on spaces of operators B(X,Y) which support only elementary surjective isometries. We also give a characterization of generalized bi-circular projections for JB * triples.

We provide a characterization of compact weighted composition operators on spaces of vector-valued Lipschitz functions. We also give estimates of the essential norm of composition operators on these spaces.

We characterize those linear projections represented as a convex combination of two surjective isometries on standard Banach spaces of continuous functions with values in a strictly convex Banach space

In this paper we establish algebraic reflexivity properties of subsets of bounded linear operators acting on spaces of vector valued Lipschitz functions. We also derive a representation for the generalized bi-circular projections on these spaces.

We study a system of differential equations in Schatten classes of operators, Cp(H)(1 £ p < ¥{\mathcal{C}_p(\mathcal{H})\,(1 \leq p < \infty}), with H{\mathcal{H}} a separable complex Hilbert space. The systems considered are infinite dimensional generalizations of mathematical models of unsupervised learning. In this new setting, we address the us...

We are interested in the isometric equivalence problem for the Cesàro operator \({C(f) (z) =\frac{1}{z} \int_{0}^{z}f(\xi) \frac{1}{1-\xi}d \xi}\) and an operator \({T_{g}(f)(z)=\frac{1}{z}\int_{0}^{z}f(\xi) g^{\prime}(\xi) d \xi}\), where g is an analytic function on the disc, on the Hardy and Bergman spaces. Then we generalize this to the isometr...

Let Ω be a connected compact Hausdorff space and X a Banach space for which the strong Banach-Stone property is valid. We give a complete characterization of the generalized bi-circular projection on the Banach spaces C(Ω) and C(Ω,X). We show that in each case generalized bi-circular projections are bi-contractive. We also give some results concern...

We characterize essential normality for certain elementary operators acting on the Hilbert-Schmidt class. We find the Aluthge transform of an elementary operator of length one. We show that the Aluthge transform of an elementary 2-isometry need not be a 2-isometry. We also characterize hermitian elementary operators of length 2.

We consider the elementary operator L, acting on the Hilbert–Schmidt Class C2(H), given by L(T)=ATB, with A and B bounded operators on H. We establish necessary and sufficient conditions on A and B for L to be a 2-isometry or a 3-isometry. We derive sufficient conditions for L to be an n-isometry. We also give several illustrative examples involvin...

We give a characterization of algebra homomorphisms and isomorphisms between spaces of Lipschitz functions with values in M n (ℂ). We show that a large class of such homomorphisms are automatically continuous. We also establish the algebraic reflexivity of the class of all algebra isomorphisms that preserve the * operation.

The isometry group of the tensor product of two symmetric sequences spaces, not isometric to a Hubert space, is algebraically reflexive provided that the tensor product supports only dyadic surjective isometries.

This paper considers a generalization of the Oja-Cox-Adams neural network model, in which an additional stimulus, not only between two given neurons but also in the two adjacent ones, may exist. In particular, the initial equation has the form dW dt=T ε CW-〈W,CW〉 * W, where C is the symmetric correlation between i and j neurons, and T ε is a tridia...

This paper provides a description of generalized bi-circular pro-jections on Banach spaces of Lipschitz functions.

We characterize the generalized bi-circular projections on various Banach spaces of both scalar and vector valued analytic functions, including the Bergman, Bloch, and Hardy spaces. We also establish that the only projections in the convex hull of two isometries on a Hardy space are generalized bi-circular projection.

We determine the essential norm of a weighted composition operator on spaces of vector valued continuous functions defined on a compact Hausdorff space. We also provide necessary and sufficient conditions for a finite sum of compact weighted composition operators to be itself a compact operator.

We establish the topological reflexivity of several spaces of analytic functions for which characterizations of their respective isometry groups are available. Namely, we consider the following spaces of analytic functions: the Novinger-Oberlin spaces consisting of those functions on the disk with the property that f′ ∈ Hp and also the more general...

We show that the isometry group of lp(X), for a separable Banach space X, is algebraically reflexive if and only if the isometry group of X is algebraically reflexive. We also show that the topological reflexivity of the isometry group of X does not imply that of lp(X). Moreover, we give a generalization of these results for a class of substitution...

Let X be a separable complex Banach space with no nontrivial -projections and not isometrically isomorphic to , where , . The space is defined to be the set of all absolutely continuous functions such that exist a.e. on and belongs to . If , the norm of f on this space is defined to be . We prove that if T is a surjective isometry T of , then T is...

We characterize norm hermitian operators on classes of tensor products of Banach spaces and derive results for the particular settings of injective and projective tensor products. We provide a characterization of bi-circular and generalized bi-circular projections on tensor products of Banach spaces supporting only dyadic surjective isometries.

The algebraic and topological reflexivity of C(X) and C(X, E) are investigated by using representations for the into isometries due to Holsztyński and Cambern.

A continuation of the authors' previous book, Isometries on Banach Spaces: Vector-valued Function Spaces and Operator Spaces, Volume Two covers much of the work that has been done on characterizing isometries on various Banach spaces. Picking up where the first volume left off, the book begins with a chapter on the Banach-Stone property. The author...

We study the existence and linear stability of stationary pulse solutions of an integro-differential equation modeling the coarse-grained averaged activity of a single layer of interconnected neurons. The neuronal connections considered feature lateral oscillations with an exponential rate of decay and variable period. We identify regions in the pa...

We consider an integro-dieren tial equation, proposed in the literature as a model of neuronal activity. We establish conditions under which an initial activity function exhibiting localized pattern formation completely characterizes the system. We also investigate how such an initial activity determines more complicated pattern formations.

We characterize generalized bi-circular projections on I(H), a minimal norm ideal of operators in B(H), where H is a separable infinite dimensional Hilbert space.

In this paper we show that the bicircular projections are precisely the Hermitian projections and prove some immediate consequences of this result.

A pair of operators on a Banach space X are isometrically equivalent if they are intertwined by a surjective isometry of X. We investigate the isometric equivalence problem for pairs of operators on specific types of Banach spaces. We study weighted shifts on symmetric sequence spaces, elementary operators acting on an ideal I of Hilbert space oper...

We study an integro-differential equation defined on a spatially extended domain, proposed in the literature as a model of short term memory. Previous researchers have established the necessary conditions for the existence of two-bump stationary solutions of this equation when considering a Heaviside firing rate function and a coupling function of...

The polar decomposition is used to analyze a class of canonical models of differential equations over the quaternions. A spectrum of oscillatory behaviors can be observed and techniques to detect directional changes of oscillations are derived. A classification of transient oscillatory behaviors encountered is also presented.

We study a system of differential equations in C(H)C(H), the space of all compact operators on a separable complex Hilbert space, HH. The systems considered are infinite-dimensional generalizations of mathematical models of learning, implementable as artificial neural networks. In this new setting, in addition to the usual questions of existence an...

We establish conditions under which Oja–Adams' learning models are gradient, semi-gradient, or gradient-like systems. We consider both single and multi-output models. The multi-output learning models are represented by matrix differential equations whose analysis require techniques from matrix calculus. We also derive the stability behavior of thes...

In this paper we consider a learning rule whose underlying space, possibly infinite dimensional, is equipped with an inner product. The rule proposed is a generalization of Oja's maximum eigenfilter algorithm. We study its convergence properties and iterative behavior. We observe a whole variety of dynamical behaviors. We establish conditions on pa...

We study the convergence behavior of a learning model with generalized Hebbian synapses.

We characterize the surjective isometries of a class of analytic functions on the disk which include the Analytic Besov spaceB p and the Dirichlet spaceD p . In the case ofB p we are able to determine the form of all linear isometries on this space. The isometries for these spaces are finite rank perturbations of integral operators. This is in c...

Let $X$ be a Banach space and let $B(X)$ denote the space of bounded operators on $X$. Two elements $S,T\inB(X)$ are isometricallyequivalent if there exists an invertible isometry $V$ such that $TV=VS$. If $X$ is a Hilbert space, then $V$ is a unitary operator and $S$ and $T$ are said to be unitarilyequivalent.

In this note we examine properties of analytic self-maps of the unit disc which induce isometries on non-Hilbertian Bergman or Hardy spaces. All isometries of these spaces are given by weighted composition operators Tf=w.fo, and it is our purpose to isolate properties of the inducing map . In both settings, maps inducing isometries are either essen...

In this paper we investigate criterion for the hyponormality, co- hyponormality, and normality of weighted composition operators acting on Hilbert spaces of vector-valued functions.

The almost transitive norm problem is studied for Lp (μ, X), C(K, X) and for certain Orlicz and Musielak-Orlicz spaces. For example if p ≠ 2 < ∞ then Lp (μ) has almost transitive norm if and only if the measure μ is homogeneous. It is shown that the only Musielak-Orlicz space with almost transitive norm is the Lp-space. Furthermore, an Orlicz space...

J. R. Holub has obtained several results for shift operators on C(X). In this paper we answer some questions of Holub and obtain extensions of many of his results. In particular, we show that C(X, R) does not admit a shift operator if X has only countably many components and each component is infinite. We show that C(X, C) does not admit a shift op...

Let (X, ∑, μ) be a σ-finite measure space and τ: X → X a measurable transformation. We give an explicit isometric isomorphism between the weighted composition operator induced by a purely dissipative transformation τ and an operator weighted shift. We use this to construct examples of subnormal w.c.o.'s. We show that if a conservative transformatio...

Let ( X , Σμ) denote a complete a-finite measure space and T : X → X a measurable ( T ⁻¹ A ε Σ each A ε Σ) point transformation from X into itself with the property that the measure μ° T ⁻¹ is absolutely continuous with respect to μ. Given any measurable, complex-valued function w ( x ) on X , and a function f in L ² (μ), define W T f ( x ) via the...

We provide corrected versions of Theorems 1 and 2 and Corollaries 3(a) and 4 of the paper mentioned in the title [this J. 32, 87-94 (1990; Zbl 0709.47027)].

Let E be a Banach sequence space with the property that if (α i ) ∈ E and |β i |≦|α i | for all i then (β i ) ∈ E and ‖(β i )‖ E ≦‖(α i )‖ E . For example E could be c o , l p or some Orlicz sequence space. If ( X n ) is a sequence of real or complex Banach spaces, then E can be used to construct a vector sequence space which we will call the E sum...

Let X be a complex Banach space. For any bounded linear operator T on X , the ( spatial ) numerical range of T is denned as the set If V ( T ) ⊆ R , then T is called hermitian . Vidav and Palmer (see Theorem 6 of [ 3 , p. 78] proved that if the set { H + iK : H and K are hermitian} contains all operators, then X is a Hilbert space. It is natural t...

Istratescu's characterization of complex strict convex (csc) Banach spaces is used to show that a modulared sum of a sequence ofcsc Banach spaces is again acsc Banach space. The equivalence of the Strong Maximum Modulus Property and complex strict convexity is used to show thatL 1(,X) iscsc whenX is (real) strictly convex and thatl 1(X n) iscsc if...

Let A denote a complex unital Banach algebra with Hermitian elements H (A). We show that if F is an analytic function from a connected open set D into A such that F(z) is normal (F(z) = u(z) + iv(z), where u(z), v(z) ∈ H(A) and u(z)v(z) = v(z)u(z)) for each z ∈ D, then F(z)F(w) = F(w)F(z) for all w, z ∈ D. This generalizes a theorem of Globevnik an...

Let l < p < ∞, p ≠ 2 and α > 0. In what follows s p ( α ) will denote the space of all real or complex sequences for which (1.1) In this paper we show that the spaces s p ( α ) are Banach spaces under the natural norm and in fact share many properties that the usual l p spaces have. Our main results give characterizations of the surjective isometri...

Let Lp(Ω, K) denote the Banach space of weakly measurable functions F defined on a finite measure space and taking values in a separable Hilbert space K for which F p = ( ∫ | F(ω) |p)1/p < + ∞. The bounded Hermitian operators on Lp(Ω, K) (in the sense of Lumer) are shown to be of the form , where B(ω) is a uniformly bounded Hermitian operator value...

Banach spaces of class g were introduced by Fleming and Jamison. This broad class includes all Banach spaces having hyperorthogonal Schauder bases and, in particular, g includes all Orlicz spaces Lø on an atomic measure space such that the characteristic functions of the atoms form a basis for Lø. The main theorem gives the structure of one paramet...

An operator A on a Banach space X is said to be adjoint abelian if there is a semi-inner product [•, •] consistent with the norm on X such that [Ax, y] = [x, Ay] for all x, y ϵX. In this paper we show that every adjoint abelian operator on C(K) or Lp(Ω, Σ, μ) (1 < p < ∞, p ≠ 2) is a multiple of an isometry whose square is the identity and hence is...

Our purpose in this paper is to give an account of the efforts of mathematicians since the appearance of Banach’s 1932 treatise to describe the isometries on a given Banach space or class of Banach spaces. Banach characterized surjective linear isometries on the continuous real-valued functions on a compact metric space as transformations of the fo...