in this paper, we introduce the concepts of $2$-isometry, collinearity, $2$%-lipschitz mapping in $2$-fuzzy $2$-normed linear spaces. also, we give anew generalization of the mazur-ulam theorem when $x$ is a $2$-fuzzy $2$%-normed linear space or $im (x)$ is a fuzzy $2$-normed linear space, thatis, the mazur-ulam theorem holds, when the $2$-isometry mapped to a $2$%-fuzzy $2$-normed linear space...

Let H be a Hilbert space with an inner product that is conjugate linear in the first variable. We do not presume unless we say so that H is separable. We denote by B(H) the set of bounded linear operators H → H. For any A ∈ B(H), A∗A is positive and one proves that it has a unique positive square root |A| ∈ B(H). We call |A| the absolute value of A. We say that U ∈ B(H) is a partial isometry if...

If R is a ring with involution, and a† is the Moore-Penrose inverse of a ∈ R, then the element a is called: EP, if aa† = a†a; partial isometry, if a∗ = a†; star-dagger, if a∗a† = a†a∗. In this paper, characterizations of partial isometries, EP elements and star-dagger elements in rings with involution are given. Thus, some well-known results are extended to more general settings.

Gau, Wang and Wu in their LAMA'2016 paper conjectured (and proved for $n\leq 4$) that an $n$-by-$n$ partial isometry cannot have a circular numerical range with non-zero center. We prove this statement holds $n=5$.

and Applied Analysis 3 The left multiplication by a ∈ A is the mapping La : A → A, which is defined as La x ax for all x ∈ A. Observe that, for a, b ∈ A, Lab LaLb and that La Lb implies a b. If a ∈ A is both Moore-Penrose and group invertible, then La† La † and La# La # in the Banach algebraL A . According to 4, Remark 12 , a necessary and sufficient condition for a ∈ A to be EP is that La ∈ L ...

We prove the equivalence of the two important facts about finite metric spaces and universal Urysohn metric spaces U, namely theorem A and theorem B below: Theorem A (Approximation): The group of isometry ISO(U) contains everywhere dense locally finite subgroup; Theorem G(Globalization): For each finite metric space F there exists another finite metric space F̄ and isometric imbedding j of F to ...

The basic notion of the article is a pair (A, U), whereA is a commutative C-algebra and U is a partial isometry such thatA ∋ a → UaU is an endomorphism of A and UU ∈ A. We give a description of the maximal ideal space of the smallest coefficient C-algebra E∗(A) of the algebra C(A, U) generated by the system (A, U).

We study the recovery conditions of weighted `1 minimization for real signal reconstruction from phaseless compressed sensing measurements when partial support information is available. A Strong Restricted Isometry Property (SRIP) condition is provided to ensure the stable recovery. Moreover, we present the weighted null space property as the sufficient and necessary condition for the success o...

This paper studies three natural pre-orders of increasing generality on the set of all completely non-unitary partial isometries with equal defect indices. We show that the problem of determining when one partial isometry is less than another with respect to these pre-orders is equivalent to the existence of a bounded (or isometric) multiplier between two natural reproducing kernel Hilbert spac...