Powers of m-isometries (original) (raw)
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Linear Algebra and its Applications, 2013
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On Stability of Isometries in Banach Spaces
Springer Optimization and Its Applications, 2011
We analyze the problem of stability of linear isometries (SLI) of Banach spaces. Stability means the existence of a function σ (ε) such that σ (ε) → 0 as ε → 0 and for any ε-isometry A of the space X (i.e., (1 − ε) x ≤ Ax ≤ (1 + ε) x for all x ∈ X) there is an isometry T such that A − T ≤ σ (ε). It is known that all finite-dimensional spaces, Hilbert space, the spaces C(K) and L p (µ) possess the SLI property. We construct examples of Banach spaces X, which have an infinitely smooth norm and are arbitrarily close to the Hilbert space, but fail to possess SLI, even for surjective operators. We also show that there are spaces that have SLI only for surjective operators. To obtain this result we find the functions σ (ε) for the spaces l 1 and l ∞. Finally, we observe some relations between the conditional number of operators and their approximation by operators of similarity.
Isometries of certain operator spaces
2004
Let X and Y be Banach spaces, and L(X, Y ) be the spaces of bounded linear operators from X into Y. In this paper we give full characterization of isometric onto operators of L(X, Y ), for a certain class of Banach spaces, that includes p , 1 < p < ∞. We also characterize the isometric onto operators of L(c 0 ) and K( 1 ), the compact operators on 1 . Furthermore, the multiplicative isometric onto operators of L( 1 ), when multiplication on L( 1 ) is taken to be the Schur product, are characterized.
Mediterranean Journal of Mathematics, 2020
We obtain the admissible sets on the unit circle to be the spectrum of a strict m-isometry on an n-finite dimensional Hilbert space. This property gives a better picture of the correct spectrum of an m-isometry. We determine that the only m-isometries on R 2 are 3-isometries and isometries giving by ±I + Q, where Q is a nilpotent operator. Moreover, on real Hilbert space, we obtain that m-isometries preserve volumes. Also we present a way to construct a strict (m + 1)-isometry with an m-isometry given, using ideas of Aleman and Suciu [7, Proposition 5.2] on infinite dimensional Hilbert space.
On the isometric conjecture of Banach
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.
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.