Linear Isometry Research Articles

Linear isometries of finite codimensions on Banach algebras of holomorphic functions

Let K be a compact subset of the complex n-space and A(K) the algebra of all continuous functions on K which are holomorphic on the interior of K. In this paper we show that under some hypotheses on K, there exists no linear isometry of finite codimension on A(K). Several compact subsets including the closure of strictly pseudoconvex domain and the product of the closure of plane domains which are bounded by a finite number of disjoint smooth curves satisfy the hypotheses. 1 Department of Mathematics, Niigata University, Niigata 950-2181, Japan. E-mail address: hatori@math.sc.niigata-u.ac.jp 2 Academic Support Center, Kogakuin University, Tokyo 192-0015, Japan. E-mail address: kt13224@ns.kogakuin.ac.jp Date: Received: 19 June 2009; Revised: 18 September 2009; Accepted: 13 October 2009. ∗ Corresponding author. 2000 Mathematics Subject Classification. Primary 46B04; Secondary 32A38, 46J10.

The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes

The main aim of the classification of linear codes is the evaluation of complete lists of representatives of the isometry classes. These classes are mostly defined with respect to linear isometry, but it is well known that there is also the more general definition of semilinear isometry taking the field automorphisms into account. This notion leads to bigger classes so the data becomes smaller. Hence we describe an algorithm that gives canonical representatives of these bigger classes by calculating a unique generator matrix to a given linear code, in a well defined manner.   &nbsp The algorithm is based on the partitioning and refinement idea which is also used to calculate the canonical labeling of a graph [12] and it similarly returns the automorphism group of the given linear code. The time needed by the implementation of the algorithm is comparable to Leon's program [10] for the calculation of the linear automorphism group of a linear code, but it additionally provides a unique representative and the automorphism group with respect to the more general notion of semilinear equivalence. The program can be used online under http://www.algorithm.uni-bayreuth.de/en/research/Coding_Theory/CanonicalForm/index.html.

On 2-local isometries on continuous vector-valued function spaces

A (not necessarily linear) mapping Φ from a Banach space X to a Banach space Y is said to be a 2 -local isometry if for any pair x , y of elements of X, there is a surjective linear isometry T : X → Y such that T x = Φ x and T y = Φ y . We show that under certain conditions on locally compact Hausdorff spaces Q, K and a Banach space E, every 2-local isometry on C 0 ( Q , E ) to C 0 ( K , E ) is linear and surjective. We also show that every 2-local isometry on ℓ p is linear and surjective for 1 ⩽ p < ∞ , p ≠ 2 , but this fails for the Hilbert space ℓ 2 .

Linear isometries between spaces of vector-valued Lipschitz functions

In this paper we state a Lipschitz version of a theorem due to Cambern concerning into linear isometries between spaces of vector-valued continuous functions and deduce a Lipschitz version of a celebrated theorem due to Jerison concerning onto linear isometries between such spaces.

The isometric extension of “into” mappings on unit spheres of AL-spaces

In this paper, we show that if V0 is an isometric mapping from the unit sphere of an AL-space onto the unit sphere of a Banach space E, then V0 can be extended to a linear isometry defined on the whole space.

On extension of isometries between unit spheres of L∞ and E

In this paper, we study the extension of isometries between the unit spheres of L ∞ and a normed space E. We find a condition under which an isometric mapping between the unit spheres can be extended to a linear isometry on the whole space L ∞.

Quaternion normed space with isometry group ℤ2

In a finite-dimensional linear space over the quaternion field, we construct a norm with the following property. Any linear isometry is one of the two transformations x ↦ x and x ↦ −x.

The Urysohn space embeds in Banach spaces in just one way

In a Master's thesis in 1985 and a subsequent paper published in 1992, the author discovered that the universal separable metric space (up to isometry) discovered by Urysohn in 1925 has a uniquely determined linear closure (up to linear isometry) when isometrically embedded in a Banach space so as to include the zero of the Banach space. The proof of this result is given in this note and the current status of some related questions is discussed.

Decoherence in Pre-symmetric Spaces

Pre-symmetric complex Banach spaces have been proposed as models for state spaces of physical systems. A structural projection on a pre-symmetric space A&#8727; represents an operation on the corresponding system, and has as its range a further pre-symmetric space which represents the state space of the resulting system and symmetries of the system are represented by elements of the group Aut(A&#8727;) of linear isometries of A&#8727;. Two structural projections R and S on the pre-symmetric space A&#8727; represent decoherent operations when their ranges are rigidly collinear. It is shown that, for decoherent elements x and y of A&#8727;, there exists an involutive element &#966;&#8727; in Aut(A&#8727;) which conjugates the structural projections corresponding to x and y, and conditions are found for &#966;&#8727; to exchange x and y. The results are used to investigate when certain subspaces of A&#8727; are the ranges of contractive projections and, therefore, represent systems arising from filtering operations.

Groups of linear isometries of spaces M q of holomorphic functions of several complex variables

C n = {z = (z1, z2, . . . , zn) | zk ∈ C, 1 ≤ k ≤ n} be the standard n-dimensional complex coordinate space. ByG we denote the ball Bn = {z ∈ C | |z1| + |z2| + · · ·+ |zn| < 1} or the polydisk U = {z ∈ C | |z1| < 1, . . . , |zn| < 1} in the space Cn, and by Γ we denote the Bergman–Shilov boundary of G: ifG = Bn, then we take Γ = Sn = {z ∈ C | |z1| + · · ·+ |zn| = 1}, and ifG = Un, then Γ = T n = {z ∈ C | |z1| = · · · = |zn| = 1}. On Γ, there exists a natural invariant probability measure σ(dζ), ζ ∈ Γ, which coincides with the normed Lebesgue measure on the sphere Sn if Γ = Sn and is the direct product of normed Lebesgue measures on the unit circles that compose the torus T n in the Cartesian product if Γ = T n. For n = 1, we have G = Bn = U = U = {z ∈ C | |z| < 1}, Γ = Sn = T n = T = {z ∈ C | |z| = 1} and σ(dζ) = |dζ|/2π, ζ ∈ T . For an arbitrary nondecreasing functions φ(t) ≥ 0 convex on [−∞,+∞), the classes φ(N) are defined (see [1, p. 91 (Russian transl.)] and [2, p. 46 (Russian transl.)]) as sets of all functions f holomorphic in the domainG and satisfying the condition

Linear isometries of Hardy spaces

According to results established by DeLeeuw-Rudin-Wermer and by Forelli, all linear isometries of any Hardy space H p (p ⩾ 1, p ≠ = 2) on the open unit disc Δ of ℂ are represented by weighted composition operators defined by inner functions on Δ. After reviewing (and completing when p = ∞) some of those results, the present report deals with a characterization of periodic and almost periodic semigroups of linear isometries of H p .

A general framework for soft-shrinkage with applications to blind deconvolution and wavelet denoising

We consider the abstract problem of approximating a function ψ 0 ∈ L 1 ( R d ) ∩ L 2 ( R d ) given only noisy data ψ δ ∈ L 2 ( R d ) . We recall that minimization of the corresponding Tikhonov functional leads to continuous soft-shrinkage and prove convergence results. If the noise-free data ψ 0 belongs to the source space L 1 − u ( R d ) ∩ L 2 ( R d ) for some 0 < u < 1 , we show convergence rates, which are order-optimal. We consider a priori parameter choice rules as well as the discrepancy principle, which is shown to be order-optimal as well. We then introduce a framework by combining soft-shrinkage with a linear invertible isometry and show that the results obtained for the abstract minimization problem can be transferred to applications such as blind deconvolution and wavelet denoising.

D-ISOMETRIC LINEAR MAPPINGS IN LINEAR d-NORMED BANACH MODULES

We prove the Hyers-Ulam stability of linear d-isometries in linear d-normed Banach modules over a unital <TEX>$C^*-algebra$</TEX> and of linear isometries in Banach modules over a unital <TEX>$C^*-algebra$</TEX>. The main purpose of this paper is to investigate d-isometric <TEX>$C^*-algebra$</TEX> isomor-phisms between linear d-normed <TEX>$C^*-algebras$</TEX> and isometric <TEX>$C^*-algebra$</TEX> isomorphisms between <TEX>$C^*-algebras$</TEX>, and d-isometric Poisson <TEX>$C^*-algebra$</TEX> isomorphisms between linear d-normed Poisson <TEX>$C^*-algebras$</TEX> and isometric Poisson <TEX>$C^*-algebra$</TEX> isomorphisms between Poisson <TEX>$C^*-algebras$</TEX>. We moreover prove the Hyers-Ulam stability of their d-isometric homomorphisms and of their isometric homomorphisms.

Error-block codes and poset metrics

Let $P = (${$1, 2,\ldots,n$}$,$≤$)$ be a poset, let $V_1, V_2,\ldots, V_n$ be a family of finite-dimensional spaces over a finite field $\mathbb F_q$ and let <br> $ V = V_1 \oplus V_2 \oplus\ldots \oplus V_n.$ <br> In this paper we endow $V$ with a poset metric such that the $P$-weight is constant on the non-null vectors of a component $V_i$, extending both the poset metric introduced by Brualdi et al. and the metric for linear error-block codes introduced by Feng et al.. We classify all poset block structures which admit the extended binary Hamming code $[8; 4; 4]$ to be a one-perfect poset block code, and present poset block structures that turn other extended Hamming codes and the extended Golay code $[24; 12; 8]$ into perfect codes. We also give a complete description of the groups of linear isometries of these metric spaces in terms of a semi-direct product, which turns out to be similar to the case of poset metric spaces. In particular, we obtain the group of linear isometries of the error-block metric spaces.

On Maps which Preserve Equality of Distance in F*-spaces

In order to generalize the results of Mazur-Ulam and Vogt, we shall prove that any map T which preserves equality of distance with T(0)=0 between two F-spaces without surjective condition is linear. Then, as a special case linear isometries are characterized through a simple property of their range.

Wold decomposition in Banach spaces

We propose a natural analog of the Wold decomposition in the case of a linear noninvertible isometry V in a Banach space X. We obtain a criterion for the existence of such a decomposition. In a reflective space, this criterion is reduced to the existence of the linear projection P: X → VX with unit norm. Separately, we discuss the problem of the Wold decomposition for the isometry V φ induced by an epimorphism φ of a compact set H in the space of continuous functions C(H). We present a detailed study of the mapping z → z m of the circle |z| = 1 with an integer m ≥ 2.

Approximately orthogonality preserving mappings on [formula omitted]-modules

We study orthogonality preserving and approximately orthogonality preserving mappings in the setting of inner product C ∗ -modules. In particular, if V and W are inner product C ∗ -modules over the C ∗ -algebra A , any scalar multiple of an A -linear isometry is an A -linear orthogonality preserving mapping. The converse does not hold in general, but it holds if A contains K ( H ) (the C ∗ -algebra of all compact operators on a Hilbert space H ). Furthermore, we give the estimate of ‖ 〈 T x , T y 〉 − ‖ T ‖ 2 〈 x , y 〉 ‖ for an A -linear approximately orthogonality preserving mapping T : V → W when V and W are inner product C ∗ -modules over a C ∗ -algebra containing K ( H ) . In the case A = K ( H ) and V, W are Hilbert, we also prove that an A -linear approximately orthogonality preserving mapping can be approximated by an A -linear orthogonality preserving mapping.

Linear isometries of some function algebras

Linear isometries of a class of logmodular algebras which are generated by unitary functions are represented by Holszty\'{n}ski type weighted composition operators. The description of these operators leads - among other things - to a characterization of the linear isometries of the disc algebra and of the Hardy space of all bounded holomorphic functions on the open unit disc of \mathbb{C} . Spectral properties of these isometries are also investigated.

Groups of linear isometries on poset structures

Let V be an n-dimensional vector space over a finite field F q and P = { 1 , 2 , … , n } a poset. We consider on V the poset-metric d P . In this paper, we give a complete description of groups of linear isometries of the metric space ( V , d P ) , for any poset-metric d P .

A Banach-Stone Theorem for completely regular spaces

In this paper, we study some Banach algebras with this property that each linear isometry between them induces a Banach algebra isometry. We obtain a Banach-Stone type theorem between Baire functions defined on completely regular spaces. As a consequence, a similar result for the space of continuous functions is deduced.