Let A and B be bounded linear operators acting on a Hilbert space H. It is shown that the triangular inequality serves as the ultimate estimate of the upper norm bound for the sum of two operators in the sense that sup{‖U∗AU + V ∗BV ‖ : U and V are unitaries} = min{‖A+ μI‖+ ‖B − μI‖ : μ ∈ C}. Consequences of the result related to spectral sets, the von Neumann inequality, and normal dilations a...

The publication of the important work Rauch and Taylor [RT75] started a hole branch research on wild perturbations Laplace-Beltrami operator. Here, we extend certain results show norm convergence resolvent. We consider (not necessarily compact) manifold with many small balls removed, number can increase as radius is shrinking, also be infinite. If distance shrinks less fast than radius, then th...

Let $ (A,| cdot |) $ be a real Banach algebra, a complex algebra $ A_mathbb{C} $ be a complexification of $ A $ and $ | | cdot | | $ be an algebra norm on  $ A_mathbb{C}  $  satisfying a simple condition together with the norm $ | cdot | $ on $ A$.  In this paper we first show that $ A^* $ is a real Banach $ A^{**}$-module if and only if $ (A_mathbb{C})^* $ is a complex Banach $ (A_mathbb{C})^{...

In this paper, we consider the Legendre spectral Galerkin and Legendre spectral collocation methods to approximate the solution of Urysohn integral equation. We prove that the approximated solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the same orders, O(n−r) in L2-norm and O(n 1 2 −r) in infinity norm, and the iterated Legendre Galerkin ...

In theoretical quantum computer science, understanding where and how computational speed-ups occur while applying quantum properties is a primary goal. In this paper, we study such problem under the framework of Quantum Query Model and prove the significance of L1-norm in the simulation of a given quantum algorithm. This result is presented by upper-bounds for the quotient between optimal class...

Given four complex matrices A, B, C and D where A ? Cn?n Cm?m given two distinct arbitrary numbers ?1 ?2, so that they are not eigenvalues of the matrix we find a nearest from set X to (with respect spectral norm) such B ! has prescribed ?2.