In this paper, first we study surjective isometries (not necessarily linear) between completely regular subspaces $A$ and $B$ of $C_0(X,E)$ and $C_0(Y,F)$ where $X$ and $Y$ are locally compact Hausdorff spaces and $E$ and $F$ are normed spaces, not assumed to be neither strictly convex nor complete. We show that for a class of normed spaces $F$ satisfying a new defined property related to their $T$-sets, such an isometry is a (generalized) weighted composition operator up to a translation. Then we apply the result to study surjective isometries between $A$ and $B$ whenever $A$ and $B$ are equipped with certain norms rather than the supremum norm. Our results unify and generalize some recent results in this context.
In this paper, we investigate the general form of surjective (not necessarily linear) isometries T : A-> B between subspaces A and B of C(X;E) and C(Y;F), respectively.
We propose a unified approach to the study of isometries on algebras of vector-valued Lipschitz maps and those of continuously differentiable maps by means of the notion of admissible quadruples. We describe isometries on function spaces of some admissible quadruples that take values in unital commutative $C^*$-algebras. As a consequence we confirm the statement of \cite[Example 8]{jp} on Lipschitz algebras and show that isometries on such algebras indeed take the canonical form.
We propose a unified approach to the study of isometries on algebras of vector-valued Lipschitz maps and those of continuously differentiable maps by means of the notion of natural $C(Y)$-valuezations that take values in unital commutative $C^*$-algebras. A precise proof of a theorem of Jarosz \cite{ja} is exhibited.
Under the right conditions on a compact metric space $X$ and on a Banach space $E$, we give a description of the $2$-local (standard) isometries on the Banach space $\hbox{Lip}(X,E)$ of vector-valued Lipschitz functions from $X$ to $E$ in terms of a generalized composition operator, and we study when every $2$-local (standard) isometry on $\hbox{Lip}(X,E)$ is both linear and surjective.
There are considered isometries on a Hilbert space. By the Wold theorem any isometry can be decomposed into a unitary operator and a unilateral shift. For a pair of isometries, even commuting, a maximal subspace reducing one isometry to a unitary operator might not reduce the other isometry. In the paper are considered pairs of commuting isometries which are completely non doubly commuting. For such pairs there are no nontrivial subspaces reducing both isometries and one of them to a unitary operator. The results describe a unitary part of an isometry in such a pair.
In the recent paper \cite{Hos}, surjective isometries, not necessarily linear, $T: {\rm AC}(X,E) \longrightarrow {\rm AC}(Y,F)$ between vector-valued absolutely continuous functions on compact subsets $X$ and $Y$ of the real line, has been described. The target spaces $E$ and $F$ are strictly convex normed spaces. In this paper, we assume that $X$ and $Y$ are compact Hausdorff spaces and $E$ and $F$ are normed spaces, which are not assumed to be strictly convex. We describe (with a short proof) surjective isometries $T: (A,\|\cdot\|_A) \longrightarrow (B,\|\cdot\|_B)$ between certain normed su…
Let $X$ be a (real or complex) rearrangement-in\-va\-riant function space on $\Om$ (where $\Om = [0,1]$ or $\Om \subseteq \bbN$) whose norm is not proportional to the $L_2$-norm. Let $H$ be a separable Hilbert space. We characterize surjective isometries of $X(H).$ We prove that if $T$ is such an isometry then there exist Borel maps $a:\Om\to\bbK$ and $σ:\Om\lra\Om$ and a strongly measurable operator map $S$ of $\Om$ into $\calB(H)$ so that for almost all $\om$ $S(\om)$ is a surjective isometry of $H$ and for any $f\in X(H)$ $$Tf(\om)=a(\om)S(\om)(f(σ(\om))) \text{ a.e.}$$ As a consequence we …
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.
In this paper, we prove that the existence of an $\varepsilon$-isometry from a separable Banach space $X$ into $Y$ (the James space or a reflexive space) implies the existence of a linear isometry from $X$ into $Y$. Then we present a set valued mapping version lemma on non-surjective $\varepsilon$-isometries of Banach spaces. Using the above results, we also discuss the rotundity and smoothness of Banach spaces under the perturbation by $\varepsilon$-isometries.
Let $Γ$ be an infinite set equipped with the discrete topology. We prove that the space $\ell_{\infty}(Γ),$ of all complex-valued bounded functions on $Γ$, satisfies the Mazur-Ulam property, that is, every surjective isometry from the unit sphere of $\ell_{\infty}(Γ)$ onto the unit sphere of an arbitrary complex Banach space $X$ admits a unique extension to a surjective real linear isometry from $\ell_{\infty}(Γ)$ to $X$.
The convex cone $SC_{\mathrm{SLip}}^1(\mathcal{X})$ of real-valued smooth semi-Lipschitz functions on a Finsler manifold $\mathcal{X}$ is an order-algebraic structure that captures both the differentiable and the quasi-metric feature of $\mathcal{X}$. In this work we show that the subset of smooth semi-Lipschitz functions of constant strictly less than $1$, denoted $SC_{1^{-}}^1(\mathcal{X})$, can be used to classify Finsler manifolds and to characterize almost isometries between them, in the lines of the classical Banach-Stone and Mykers-Nakai theorems.
A well-known theorem of Paulsen says that if $\mathcal{A}$ is a unital operator algebra and $ϕ:\mathcal{A}\to B(\mathcal{H})$ is a unital completely bounded homomorphism, then $ϕ$ is similar to a completely contractive map $ϕ'$. Motivated by classification problems for Hilbert space contractions, we are interested in making the inverse $ϕ'^{-1}$ completely contractive as well whenever the map $ϕ$ has a completely bounded inverse. We show that there exist invertible operators $X$ and $Y$ such that the map $$ XaX^{-1}\mapsto Yϕ(a)Y^{-1} $$ is completely contractive and is "almost" isometric on a…
We investigate the structure of norm-preserving and linear but not necessarily surjective operators on variable-exponent, discrete Lebesgue spaces. A certain class of isometries, novel to this work, are especially considered; this class completely coincides with all isometries when the Lebesgue space is classical, i.e. of a fixed-exponent. For said isometries it is shown that their actions are completely determined by pairs consisting of set-mappings and bounded functions on $\mathbb{N}$. This result recovers the previously-known structure of isometries on fixed-exponent spaces as a special ca…
This paper studies three natural pre-orders of increasing generality on the set of all completely non-unitary partial isometries with equal defect indices. We show that the problem of determining when one partial isometry is less than another with respect to these pre-orders is equivalent to the existence of a bounded (or isometric) multiplier between two natural reproducing kernel Hilbert spaces of analytic functions. For large classes of partial isometries these spaces can be realized as the well-known model subspaces and deBranges-Rovnyak spaces. This characterization is applied to investig…
We prove that for a bijective, unital, linear map between absolute order unit spaces is an isometry if, and only if, it is absolute value preserving. We deduce that, on (unital) $JB$-algebras, such maps are precisely Jordan isomorphisms. Next, we introduce the notions of absolutely matrix ordered spaces and absolute matrix order unit spaces and prove that for a bijective, unital, linear map between absolute matrix order unit spaces is a complete isometry if, and only if, it is completely absolute value preserving. We obtain that on (unital) C$^*$-algebras such maps are precisely C$^*$-algebra …
A family $\{T_j\}_{j\in J}$ of commuting Hilbert space operators is said to be a spherical isometry if $\sum_{j\in J}T^*_jT_j=1$ in the weak operator topology. We show that every commuting family $\Cal F$ of spherical isometries has a commuting normal extension $\hat{\Cal F}$. Moreover, if $\hat{\Cal F}$ is minimal, then there exists a natural short exact sequence $0\to\Cal C\to C^*(\Cal F)\to C^*(\hat{\Cal F})\to 0$ with a completely isometric cross-section, where $\Cal C$ is the commutator ideal in $C^*(\Cal F)$. We also show that the space of Toeplitz operators associated to $\Cal F$ is com…
It is shown that every linear surjective isometry between two right, full, Hilbert C*-modules is a sum of two maps : a (bi-) module map (which is completely isometric and preserves the inner product) and a map that reverses the (bi-) module actions. Every such isometry can be extended to an isometry of the C* linking algebras.
Let $E$ be an operator space in the sense of the theory recently developed by Blecher-Paulsen and Effros-Ruan. We introduce a notion of $E$-valued non commutative $L_p$-space for $1 \leq p < \infty$ and we prove that the resulting operator space satisfies the natural properties to be expected with respect to e.g. duality and interpolation. This notion leads to the definition of a ``completely p-summing" map which is the operator space analogue of the $p$-absolutely summing maps in the sense of Pietsch-Kwapień. These notions extend the particular case $p=1$ which was previously studied by Effro…
We study self-adjoint semigroups of partial isometries on a Hilbert space. These semigroups coincide precisely with faithful representations of abstract inverse semigroups. Groups of unitary operators are specialized examples of self-adjoint semigroups of partial isometries. We obtain a general structure result showing that every self-adjoint semigroup of partial isometries consists of "generalized weighted composition" operators on a space of square-integrable Hilbert-space valued functions. If the semigroup is irreducible and contains a compact operator then the underlying measure space is p…
Botelho, Jamison, and Moln\' ar have recently described the general form of surjective isometries of Grassmann spaces on complex Hilbert spaces under certain dimensionality assumptions. In this paper we provide a new approach to this problem which enables us first, to give a shorter proof and second, to remove dimensionality constraints completely. In one of the low dimensional cases, which was not covered by Botelho, Jamison, and Moln\' ar, an exceptional possibility occurs. As a byproduct, we are able to handle the real case as well. Furthermore, in finite dimensions we remove the surectivit…
We prove some noncommutative analogues of a theorem by Plotkin and Rudin about isometries between subspaces of Lp-spaces. Let 0<p<\infty, p not an even integer. The main result of this paper states that in the category of unital subspaces of noncommutative probability Lp-spaces, under some boundedness condition, the unital completely isometric maps come from *-isomorphisms of the underlying von Neumann algebras. Some applications are given, including to non commutative H^p spaces.
We will give an outline of the main results in our recent AMS Memoir, and include some new results, exposition and open problems. In that memoir we developed a general dilation theory for operator valued measures acting on Banach spaces where operator-valued measures (or maps) are not necessarily completely bounded. The main results state that any operator-valued measure, not necessarily completely bounded, always has a dilation to a projection-valued measure acting on a Banach space, and every bounded linear map, again not necessarily completely bounded, on a Banach algebra has a bounded homo…
For realcompact spaces X and Y we give a complete description of the linear biseparating maps between spaces of vector-valued continuous functions on X and Y, where special attention is paid to spaces of vector-valued bounded continuous functions. These results are applied to describe the linear isometries between spaces of vector-valued bounded continuous and uniformly continuous functions.
In this paper we propose a different (and equivalent) norm on $S^{2} ({\mathbb{D}})$ which consists of functions whose derivatives are in the Hardy space of unit disk. The reproducing kernel of $S^{2}({\mathbb{D}})$ in this norm admits an explicit form, and it is a complete Nevanlinna-Pick kernel. Furthermore, there is a surprising connection of this norm with $3$ -isometries. We then study composition and multiplication operators on this space. Specifically, we obtain an upper bound for the norm of $C_φ$ for a class of composition operators. We completely characterize multiplication operators…
A bounded linear Hilbert space operator $S$ is said to be a $2$-isometry if the operator $S$ and its adjoint $S^*$ satisfy the relation $S^{*2}S^{2} - 2 S^{*}S + I = 0$. In this paper, we study Hilbert space operators having liftings or dilations to $2$-isometries. The adjoint of an operator which admits such liftings is characterized as the restriction of a backward shift on a Hilbert space of vector-valued analytic functions. These results are applied to concave operators (i.e., operators $S$ such that $S^{*2}S^{2} - 2 S^{*}S + I \le 0$) and to operators similar to contractions or isometries…
In this paper, we introduce a unitary invariant $Γ$ defined on the unit ball of $B(H)^n$ in terms of the characteristic function, the noncommutative Poisson kernel, and the defect operator associated with a row contraction. We show that $Γ$ detects the pure row isometries and completely classify them up to a unitary equivalence. We also show that $Γ$ detects the pure row contractions with polynomial characteristic functions and completely non-coisometric row contractions. In particular, we show that any completely non-coisometric row contraction with constant characteristic function is homogen…
We prove that in the setting of operator spaces the result of Davis, Figiel, Johnson and Pelczynski on factoring weakly compact operators holds accordingly. Though not related directly to the main theorem we add a remark on the description of weakly compact subsets in the dual of noncommutative vector valued ${\rm L}_1$.
In this article, we characterize nearly invariant subspaces of finite defect for the backward shift operator acting on the vector-valued Hardy space which is a vectorial generalization of a result of Chalendar-Gallardo-Partington (C-G-P). Using this characterization of nearly invariant subspace under the backward shift we completely describe the almost invariant subspaces for the shift and its adjoint acting on the vector valued Hardy space.
The Cauchy dual subnormality problem (for short, CDSP) asks whether the Cauchy dual of a $2$-isometry is subnormal. In this paper, we address this problem for cyclic $2$-isometries. In view of some recent developments in operator theory on function spaces (see \cite{AM, LGR}), one may recast CDSP as the problem of subnormality of the Cauchy dual $\mathscr M'_z$ of the multiplication operator $\mathscr M_z$ acting on a de Branges-Rovnyak space $\mathcal H(B),$ where $B$ is a vector-valued rational function. The main result of this paper characterizes the subnormality of $\mathscr M'_z$ on $\mat…
We present a completely new structure theoretic approach to the dilation theory of linear operators. Our main result is the following theorem: if $X$ is a super-reflexive Banach space and $T$ is contained in the weakly closed convex hull of all invertible isometries on $X$, then $T$ admits a dilation to an invertible isometry on a Banach space $Y$ with the same regularity as $X$. The classical dilation theorems of Sz.-Nagy and Akcoglu-Sucheston are easy consequences of our general theory.
Let $Ω$ be a compact Hausdorff space, let $E$ be a Banach space, and let $C(Ω, E)$ stand for the Banach space of all $E$-valued continuous functions on $Ω$ under supnorm. In this paper we study when nuclear operators on $C(Ω, E)$ spaces can be completely characterized in terms of properties of their representing vector measures. We also show that if $F$ is a Banach space and if $T:\ C(Ω, E)\rightarrow F$ is a nuclear operator, then $T$ induces a bounded linear operator $T^\#$ from the space $C(Ω)$ of scalar valued continuous functions on $Ω$ into $\slN(E,F)$ the space of nuclear operators from…
Let T and C be two Hilbert space operators. We prove that if T is near, in a certain sense, to an operator completely polynomially dominated with a finite bound by C, then T is similar to an operator which is completely polynomially dominated by the direct sum of C and a suitable weighted unilateral shift. Among the applications, a refined Banach space version of Rota similarity theorem is given and partial answers to a problem of K. Davidson and V. Paulsen are obtained. The latter problem concerns CAR-valued Foguel-Hankel operators which are generalizations of the operator considered by G. Pi…
We extend some results about sampling of entire functions of exponential type to Banach spaces. By using generator $D$ of one-parameter group $e^{tD}$ of isometries of a Banach space $E$ we introduce Bernstein subspaces $\mathbf{B}_σ(D),\>\>σ>0,$ of vectors $f$ in $E$ for which trajectories $e^{tD}f$ are abstract-valued functions of exponential type which are bounded on the real line. This property allows to reduce sampling problems for $e^{tD}f$ with $f\in \mathbf{B}_σ(D)$ to known sampling results for regular functions of exponential type $σ$.
We introduce a preorder relation in the collection of all operator valued completely positive maps on a full Hilbert C*-module and characterize this relation in terms of the Stinespring construction associated to each completely positive map.
We introduce a method to construct general multivariate positive definite kernels on a nonempty set $X$ that employs a prescribed bounded completely monotone function and special multivariate functions on $X$.\ The method is consistent with a generalized version of Aitken's integral formula for Gaussians.\ In the case where $X$ is a cartesian product, the method produces nonseparable positive definite kernels that may be useful in multivariate interpolation.\ In addition, it can be interpreted as an abstract multivariate generalization of the well-established Gneiting's model for constructing …
We define a scale of L^q Carleson norms, all of which characterize the membership of a function in BMO. The phenomenon is analogous to the John-Nirenberg inequality, but on the level of Carleson measures. The classical Carleson condition corresponds to the L^2 case in our theory. The result is applied to give a new proof for the L^p-boundedness of paraproducts with a BMO symbol. A novel feature of the argument is that all p are covered at once in a completely interpolation-free manner. This is achieved by using the L^1 Carleson norm, and indicates the usefulness of this notion. Our approach is…
We study Hausdorff-Young type inequalities for vector-valued Dirichlet series which allow to compare the norm of a Dirichlet series in the Hardy space $\mathcal{H}_{p} (X)$ with the $q$-norm of its coefficients. In order to obtain inequalities completely analogous to the scalar case, a Banach space must satisfy the restrictive notion of Fourier type/cotype. We show that variants of these inequalities hold for the much broader range of spaces enjoying type/cotype. We also consider Hausdorff-Young type inequalities for functions defined on the infinite torus $\mathbb{T}^{\infty}$ or the boolean …
Let U be an open subset of R^n. Let L^2=L^2(U,dx) and H^1_0=H^1_0(U) be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group G of invertible operators on H^1_0 which preserve the L^2-inner product. When U is bounded and the border $\partial U$ is smooth, this group acts as the intertwiner of the H^1_0 solutions of the non-homogeneous Helmholtz equation $u-Δu=f$, $u|_{\partial U}=0$. We show that G is a real Banach-Lie group, whose Lie algebra is (i times) the space of symmetrizable operators. We discuss the spectrum of operators belo…
In this article, we introduce local completely positive $k$-linear maps between locally $C^{\ast}$-algebras and obtain Stinespring type representation by adopting the notion of "invariance" defined by J. Heo for $k$-linear maps between $C^{\ast}$-algebras. Also, we supply the minimality condition to make certain that minimal representation is unique up to unitary equivalence. As a consequence, we prove Radon-Nikodým theorem for unbounded operator-valued local completely positive invariant $k$-linear maps. The obtained Radon-Nikodým derivative is a positive contraction on some Hilbert space wit…
