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In this paper, we investigate the properties of the reduced minimum modulus in the context of Banach spaces. Given a Banach space $X$, we denote the algebra of bounded operators on $X$ as $B(X)$. Our primary focus is on examining the relationship between the reduced minimum modulus of a given operator $T \in B(X)$ and its associated left and right multiplication operators, denoted by $L_T: S \mapsto TS$ and $R_T: S \mapsto ST$, respectively. By analyzing these relationships, we present a comprehensive analysis of their properties and derive novel results concerning the reduced minimum modulus of $L_T$ and $R_T$.

We give several basic and spectral properties of classes of closed n-paranormal and closed n ∗ -paranormal operators on dense domains in complex separable Hilbert spaces. We prove that for both of these classes of operators, the null space of (T - μ I) and the range of R (E μ) are identical, where E μ is the Riesz projection with respect to an isolated point μ of the spectrum. We show that they satisfy Weyl's theorem. Certain properties related to the reduced minimum modulus are also established. [ABSTRACT FROM AUTHOR]

In this paper, we define the notions of semi-regular operator, analytical core, surjectivity modulus and the injectivity modulus of bounded linear operators on non-Archimedean Banach spaces over $\mathbb{K}.$ We give a necessary and sufficient condition on the range of bounded linear operators to be closed. Moreover, many results are proved.

Suppose $\mathfrak{B}(H)$ is the Banach algebra of all bounded linear operators on a Hilbert space $H$ with $\dim(H)\geq 3$. Let $\gamma(.)$ denote the reduced minimum modulus of an operator. We charaterize surjective maps $\varphi$ on $\mathfrak{B}(H)$ satisfying $$\gamma(\varphi(T)\varphi(S))=\gamma(T S)\;\;\;(T, S\in \mathfrak{B}(H)).$$ Also, we give the general form of surjective maps on $\mathfrak B(H)$ preserving the reduced minimum modulus of Jordan triple products of operators.

For a closed linear relation everywhere defined on a Hilbert space the concepts of isometry, co-isometry, partial isometry, and generalized inverse are introduced and studied. Part of the results proved in this paper improve and generalize some results known for these concepts. In particular, we extend those of [Acta Sci. Math. (Szeged), 70 (2004), 767–781] and [Studia Math. 205 (2011), no. 1, 71–82]. [ABSTRACT FROM AUTHOR]

Let H be a complex Hilbert space and let B (H) be the algebra of all bounded linear operators on H. In this paper, for T ∈ B (H) and a unit vector x ∈ H , we introduce a local version of the reduced minimum modulus of T at x, noted by γ (T , x) . Properties of this quantity are investigated. We study the relations between γ (T , x) and the Moore–Penrose inverse, spectrum of | T | and the local spectrum of | T | at x. At the end of this paper we will be interested in several problems around this quantity (preserving, continuity, local spectral theory). [ABSTRACT FROM AUTHOR]

The concept of the Drazin inverse of multivalued operators in a Banach space studied by A. Ghorbel and M. Mnif [Monatsh Math. 2019;189:273–293] is generalized in the context of the generalized Drazin inverse of multivalued operators [Rocky Mountain J Math. 2020;50(4):1387–1408]. The purpose of this paper is to present new additive results for this concept. In particular, we give a sufficient condition for an everywhere defined linear relation to have at most one Drazin inverse. Some properties and the explicit expressions for the Drazin inverse of the product are obtained. Also, some results of C. Deng, H. Du [Proc Amer Math Soc. 2006;134:3309–3317] concerning the reduced minimum modulus of Drazin inverses of linear operators on Hilbert spaces are extended to the case of linear relations. [ABSTRACT FROM AUTHOR]

Let H1, H2 be complex Hilbert spaces and T be a densely defined closed linear operator from its domain D(T), a dense subspace of H1, into H2. Let N(T) denote the null space of T and R(T) denote the range of T. Recall that C(T):= D(T) ∩ N(T)⊥ is called the carrier space of T and the reduced minimum modulus γ(T) of T is defined as: γ (T) : = inf { ‖ T (x) ‖ : x ∈ C (T) , ‖ x ‖ = 1 }. Further, we say that T attains its reduced minimum modulus if there exists x0 ∈ C(T) such that ∥x0∥ = 1 and ∥T(x0)∥ = γ(T). We discuss some properties of operators that attain reduced minimum modulus. In particular, the following results are proved. The operator T attains its reduced minimum modulus if and only if its Moore-Penrose inverse T† is bounded and attains its norm, that is, there exists y0 ∈ H2 such that ∥y0∥ = 1 and ∥T†∥ = ∥T†(y0)∥. For each ϵ > 0, there exists a bounded operator S such that ∥S∥ ≤ ϵ and T + S attains its reduced minimum. [ABSTRACT FROM AUTHOR]

For a closed linear relation in a Hilbert space the notions of minimum modulus, essential g-ascent, essential ascent and essential descent are introduced and studied. We prove that some results of E. Chafai and M. Mnif [3] related to the stability of the essential descent and descent of a linear relation T everywhere defined such that $${T(0)\subseteq \mathsf{ker}(T)}$$ by a finite rank operator F commuting with T, remain valid when F is an everywhere defined linear relation and without the assumption that $${T(0)\subseteq \mathsf{ker}(T)}$$ . We studied also the stability of the essential g-ascent and the essential ascent under a finite rank relation. Motivated by the recent work of T. Álvarez and A. Sandovici [1], we extend to a closed linear relation, the well known notion of minimum modulus of a linear operator (H. A. Gindler and A. E. Taylor [7]). Also, we introduce and study the new notion of minimum g-modulus for a linear relation. [ABSTRACT FROM AUTHOR]

We introduce and study the class of extremally rich JB*-triples. We establish new results to determine the distance from an element a in an extremally rich JB*-triple E to the set ∂e(E1) of all extreme points of the closed unit ball of E. More concretely, we prove that dist(a,∂e(E1))= max {1,‖α‖-1} for every α ϵ E which is not Brown-Pedersen quasi-invertible. As a consequence, we determine the form of the λ-function of Aron and Lohman on the open unit ball of an extremally rich JB*-triple E by showing that λ(a) = 1=2 for every non-BP quasi-invertible element a in the open unit ball of E. We also prove that for an extremally rich JB*-triple E, the quadratic conorm γq(·) is continuous at a point α ϵ E if and only if either a is not von Neumann regular (i.e., γq(a) = 0) or a is Brown-Pedersen quasi-invertible. [ABSTRACT FROM AUTHOR]

We prove that if φ : M n ( C ) → M n ( C ) is a bijective map which preserves the reduced minimum modulus on differences of matrices, then φ is automatically bicontinuous and therefore, by a result of Mashreghi and Stepanyan, of a standard form. [ABSTRACT FROM AUTHOR]

Let X and Y be infinite-dimensional complex Banach spaces, and let B ( X ) (resp. B ( Y ) ) denote the algebra of all bounded linear operators on X (resp. on Y ). We describe bijective bicontinuous maps φ from B ( X ) to B ( Y ) satisfying γ ( φ ( S ± φ ( T ) ) ) = γ ( S ± T ) for all S , T ∈ B ( X ) , where γ ( T ) is the reduced minimum modulus of an operator T . An analogue result for the finite-dimensional case is obtained. [ABSTRACT FROM AUTHOR]

In this paper, the stabilities of the nullity, invertibility and generalized invertibility of linear combinationsorare investigated, whereandbe twoandpotents, respectively. [ABSTRACT FROM PUBLISHER]

A classical result of Apostol (Michigan Math. J. 32, 279-294, ) concerning the reduced minimum modulus of bounded linear operators acting in Hilbert spaces is extended to the case of linear relations. [ABSTRACT FROM AUTHOR]

Abstract: We describe linear maps from a -algebra onto another one preserving different spectral quantities such as the minimum modulus, the surjectivity modulus, and the reduced minimum modulus. [Copyright &y& Elsevier]

Let H1,H2 be Hilbert spaces and T be a closed linear operator defined on a dense subspace D(T ) in H1 and taking values in H2. In this article we prove the following results: (i) Range of T is closed if and only if 0 is not an accumulation point of the spectrum σ(T *T ) of T *T , In addition, if H1 = H2 and T is self-adjoint, then (ii) inf {∥T x∥: x ∈ D(T ) ∩ N(T )⊥∥x;∥ = 1} = inf {λ∣: 0 ≠= λ ∈ σ(T )}, (iii) Every isolated spectral value of T is an eigenvalue of T , (iv) Range of T is closed if and only if 0 is not an accumulation point of the spectrum σ(T ) of T , (v) σ(T ) bounded implies T is bounded. We prove all the above results without using the spectral theorem. Also, we give examples to illustrate all the above results. [ABSTRACT FROM AUTHOR]

Let $${\mathcal{B}}(\rm X)$$ be the algebra of all bounded linear operators on a complex Banach space X and γ( T) be the reduced minimum modulus of operator $$T \in {\mathcal{B}}(\rm X)$$ . In this work, we prove that if $$\phi : {\mathcal{B}}(\rm X) \rightarrow {\mathcal{B}}(\rm X)$$ , is a surjective linear map such that $$\phi(I)$$ is an invertible operator, then $$\gamma(T) = \gamma(\phi(T))$$ , for every $$T \in {\mathcal{B}}(\rm X)$$ , if and only if, either there exist two bijective isometries $$U_1 \in {\mathcal{B}}(\rm X)$$ and $$U_2 \in {\mathcal{B}}(\rm X)$$ such that $$\phi(T) = U_{1}TU_{2}$$ for every $$T \in {\mathcal{B}}(\rm X)$$ , or there exist two bijective isometries $$V_{1} \in {\mathcal{B}}(\rm{X}^\prime ,\rm{X})$$ and $$V_{2} \in {\mathcal{B}}(\rm{X},\rm{X}^\prime)$$ such that $$\phi(T) = V_{1}T^\prime V_{2}$$ for every $$T \in {\mathcal{B}}(\rm{X})$$ . This generalizes for a Banach space the Mbekhta’s theorem [12]. [ABSTRACT FROM AUTHOR]

We define the reduced minimum modulus $$\gamma _{\mathcal {A}} (a)$$ of a nonzero element a in a unital C*-algebra $${\mathcal{A}}$$ by $$\gamma _{\mathcal {A}} (a) = \rm{inf}\{\parallel a-b \parallel | \rm{AL}({\it a})\subsetneqq AL({\it b}), b \in {\mathcal {A}}\}$$ . We prove that $$\gamma _{\mathcal {A}} (a) = \rm{inf}\{\lambda | \lambda \in \sigma((a^{*}a)^{\frac{1}{2}})\setminus \{0\}\}$$ . Applying this result to $${\mathcal{A}}$$ and its closed two side ideal $${\mathcal{I}}$$ , we get that dist $$(a, \Phi^{c}_{l}({\mathcal {A}})) = \rm{min}\{\lambda | \lambda \in \sigma(\pi((a^{*}a)^{\frac{1}{2}}))\}$$ , $$\forall a \in {\mathcal {A}}\setminus \{0\}$$ and $$\gamma _{\mathcal {B}}(\pi(a)) = \text{sup}\{\gamma _{\mathcal {A}}(a + k) | k \in {\mathcal {I}}\}$$ for any $$a \in {\mathcal{A}}\setminus{\mathcal{I}}$$ if RR $$({\mathcal{A}})$$ = 0, where $${\mathcal{B}} = {\mathcal{A}}/{\mathcal{I}}$$ and $$\pi : {\mathcal{A}} \rightarrow {\mathcal{B}}$$ is the quotient homomorphism and $$\Phi^{c}_{l} ({\mathcal{A}}) =\{a \in {\mathcal {A}} | \pi(a) \text{is not left invertible in}\,{\mathcal {B}}\}$$ . These results generalize corresponding results in Hilbert spaces. [ABSTRACT FROM AUTHOR]

Let X and Y be two Banach spaces over the complex field C and let T:X→Y be a bounded linear operator with the generalized inverse T+. Let T=T+δT be a bounded linear operator with &par;T+&par;&par;δT&par;<1/2. Let γ(T) denote the reduced minimum modulus of T. We first establish the upper semi-continuity theorem for γ(T). Furthermore, if dim Ker T=dim Ker T<+∞ or R(T)∩Ker T+=0, then γ(T) is continuous under the operator norm. Also, we have a corollary in Hilbert spaces, which improves the result in [Linear Algebra Appl., 262 (1997) 229–242, Theorem 4.1]. [Copyright &y& Elsevier]

Let X and Y be two Banach spaces, and let T:X→Y be a bounded linear operator. We study the perturbation problem of the reduced minimum modulus of T, and in particular we correct a mistake in a recent paper. [Copyright &y& Elsevier]

In this paper we compute the distance between the connected components of the sets of all partial isometries and essential partial isometries. Some factorization results are also proved. This complements previous results due to P.R. Halmos and J.E. McLaughlin [ABSTRACT FROM AUTHOR]