Powers of m-isometries (original) (raw)

Geometry & Topology, 2021

Let V be a Banach space all of whose n-dimensional subspaces are isometric for some fixed n, 1 < n < dim(V). In 1932, Banach asked if under this hypothesis V is necessarily a Hilbert space. The question has been answered positively by various authors (most recently by Gromov in 1967) for even n and all V , and for odd n and large enough dim(V). In this paper we give a positive answer for real V and odd n of the form n = 4k + 1, n = 133. Our proof relies on a new characterization of ellipsoids in R n , n ≥ 5, as the only symmetric convex bodies all of whose linear hyperplane sections are linearly equivalent affine bodies of revolution.

Strict isometries of arbitrary orders

Linear Algebra and its Applications, 2012

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 Hilbert space H then {I, A * A, (A *) 2 A 2 ,. .. , (A *) n−1 A n−1 } is a linearly independent set of operators. This result allows to extend further the isometric interdependence of L and its symbols. In particular we show that if L is a p-isometry then A is a strict p − 1-(or p − 2-)isometry if and only if B * is a strict 2-(or 3-)isometry.

A Note on Weak Stability of 𝜺-Isometries on Certain Banach Spaces

Journal of Science and Arts

In this paper, we will discuss the weak stability of ε-isometries on certain Banach spaces. Let f: X → Y be a standard ε-isometry. If Y^* is strictly convex, then for any x^*∈X^*, there is φ∈Y^* that satisfies ‖φ‖ ≡r=‖x^* ‖, such that |〈x^*,x〉-〈φ,f(x)〉|≤2rε,x∈X. Also, we show that if X and Y are both L_P spaces (1

On isometric reflexions in Banach spaces

We obtain the following characterization of Hilbert spaces. Let E be a Banach space whose unit sphere S has a hyperplane of symmetry. Then E is a Hilbert space iff any of the following two conditions is fulfilled: a) the isometry group Iso E of E has a dense orbit in S; b) the identity component G 0 of the group Iso E endowed with the strong operator topology acts topologically irreducible on E. Some related results on infinite dimentional Coxeter groups generated by isometric reflexions are given which allow to analyse the structure of isometry groups containing sufficiently many reflexions.

On Isometric Embedding ℓ_p^m→ S_∞^n and Unique operator space structure

2019

We study existence of isometric embedding of ℓ_p^m into S_∞^n for 1≤ p,q≤∞. For p∈(2,∞)∪{1}, we show that indeed ℓ_p^2 does not embed isomerically into S_∞. This verifies a guess of Pisier and generalizes the main result of <cit.> in more generality. We also show that S_1^m does not embed isometrically into S_p^n for all 1<p<∞ and m≥ 2. As a consequence, we establish noncommutative analogue of some of the results in <cit.>. Applying this, we show that (C^2,._B_p,q) does not embed isometrically into S_∞ for 2<p,q<∞. We also prove (C^2,._B_p,q) does not have a unique operator space structure whenever (p,q)∈(1,∞)×[1,∞)∪[1,∞)×(1,∞) by showing that they do not have Property P or two summing property. This produces genuinely new examples of two dimensional Banach spaces without unique operator space structure, providing a partial answer to a question of Paulsen. The main ingredients in our proof are notion of Birkhoff-James orthogonality and norm parallelism for op...

Classes of operators related to m-isometric operators

Operators and Matrices

Isometries played a pivotal role in the development of operator theory, in particular with the theory of contractions and polar decompositions and has been widely studied due to its fundamental importance in the theory of stochastic processes, the intrinsic problem of modeling the general contractive operator via its isometric dilation and many other areas in applied mathematics. In this paper we present some properties of n-quasi-(m, C)-isometric operators. We show that a power of a n-quasi-(m, C)-isometric operator is again a n-quasi-(m, C)-isometric operator and some products and tensor products of n-quasi-(m, C)-isometries are again n-quasi-(m, C)-isometries.

Non-negative Integer Power of a Hyponormal m-isometry is Reﬂexive

Asian Research Journal of Mathematics

Sarason did pioneer work on reflexive operator and reflexivity of normal operators, however, he did not used the word reflexive but his results are equivalent to say that every normal operator is reflexive. The word reflexive was suggested by HALMOS and first appeared in H. Rajdavi and P. Rosenthals book `Invariant Subspaces’ in 1973. This line of research was continued by Deddens who showed that every isometry in B(H) is reflexive. R. Wogen has proved that `every quasi-normal operator is reflexive’. These results of Deddens, Sarason, Wogen are particular cases of theorem of Olin and Thomson which says that all sub-normal operators are reflexive. In other direction, Deddens and Fillmore characterized these operators acting on a finite dimensional space are reflexive. J. B. Conway and Dudziak generalized the result of reflexivity of normal, quasi-normal, sub-normal operators by proving the reflexivity of Vonneumann operators. In this paper we shall discuss the condition under which m...