The distance between certainn-dimensional Banach spaces

Vietnam Journal of Mathematics, 2019

In this article, we estimate the spectral radius of linear bounded operators on Banach spaces when they are subjected to perturbations of the single or multi-block structured form. Then, we derive formulae and bounds for structured distance to non-invertibility (resp. non-left, non-right invertible operator) of an invertible (resp. left, right invertible) operator on Banach spaces. Some special cases of that estimate will also be considered.

Integral Equations and Operator Theory, 1992

If P is a positive operator on a Hilbert space H whose range is dense, then a theorem of Foias, Ong, and Rosenthal says that: II[qo(P)]-lT[tp(P)]ll < 12 max{llTII, IIp-1TPII} for any bounded operator T on H, where q~ is a continuous, concave, nonnegative, nondecreasing function on [0, IIPII]. This inequality is extended to the class of normal operators with dense range to obtain the inequality II[tp(N)]-lT[tp(N)]ll < 12c 2 max{llTII, IIN-ITNII} where tp is a complex valued function in a class of functions called vase-like, and c is a constant which is associated with q~ by the definition of vase-like. As a corollary, it is shown that the reflexive lattice of operator ranges generated by the range NH of a normal operator N consists of the ranges of all operators of the form tp(N), where q0 is vase-like. Similar results are obtained for scalar-type spectral operators on a Hilbert space,

Matemática Contemporânea, 2008

Let X be a Banach space and L(X) be the Banach algebra of bounded operators on X. In this note we prove that if we have a compact subset K of a commutative sub-algebra of L(X), and given ε > 0, then it is possible to define a new norm in X, equivalent to its given norm, in such a way that inside a neighborhood Uε of this compact set in the subalgebra, the norms of all the operators differ from their spectral radius in less than ε. If X is a Hilbert space then it is possible to define this new norm as an Hilbertian norm. When one wants to estimate the norms of functions of bounded operators the properties of norms on L(X) usually are very helpful. But imposing assumptions on the spectral radius of an operator is better than imposing on its norm, because the first is intrinsic and the second one is not. But for certain sets of bounded operators one can change the norm of the space to an equivalent norm in such a way that the norms of the bounded operators are as near as we wish to their spectral radius. This fact has been widely used by many authors, including C. Pugh classical paper [1], for example. In our previous papers Rodrigues-Sola Morales [2] and [3] we also used this fact, and in [2] we proved that if we have a finite family of bounded linear operators T n , such that T n commutes with T m for every m, n, then it is possible to define a new norm on the Banach space X, equivalent to the initial norm,

Journal of Approximation Theory, 1999

Studia Mathematica, 2006

A Banach space X is asymptotically symmetric (a.s.) if for some C < ∞, for all m ∈ N, for all bounded sequences (x i j ) ∞ j=1 ⊆ X, 1 ≤ i ≤ m, for all permutations σ of {1, . . . , m} and all ultrafilters U1, . . . , Um on N,

m-hikari.com

In this paper we find new estimate for the numerical radius for sums and products of Hilbert space operators. Also we generalized and sharpened some recent inequalities for the numerical radius.

Pacific Journal of Mathematics, 1969

Journal of Computational and Applied Mathematics, 1997

By using a norm generated by the error series of a sequence of interpolation polynomials, we obtain in this paper ~ertain Banach spaces. A relation between these spaces and the space (Co, S) with norm generated by the error series of the best polynomial approximations (minimax series) is established.

Advances in operator theory, 2019

In this paper, we generalize some inequalities for the approximation numbers of an element in a normed (Banach) algebra X and, as an application, we present inequalities for the quasinorms of some ideals defined by means of the approximation numbers.

Numerical Functional Analysis and Optimization, 2015