Let 𝒜 be a complex unital commutative Banach algebra. Let φ𝒜→ℂ be a map such that for x,y∈𝒜 , φ(x)-φ(y)∈σ_ε(x-y) and φ has ℂ -linear differentials almost everywhere. Then φ is approximately multiplicative. A similar conclusion is reached by replacing the differential condition with comparable assumptions on the map. This result is similar to the Kowalski–Słodkowski theorem. Analogous versions of it are also discussed for the exponential spectrum and for a particular class of the Ransford spectrum.

The restricted topology is defined to investigate sets between the spectrum and its hull that arise by filling the holes. In this paper, we study this topology in the context of Banach algebras. We explore the properties of the associated connected hull that are in common or in contrast with the well-known notions of spectrum and exponential spectrum. Further, we deduce sufficient conditions for the commutativity of the connected hull.

Motivated by the study of multiplicative linear functionals in reproducing kernel Hilbert space (RKHS) with normalized complete Pick kernel, we define and study the multiplicative linear map between two RKHS. We identify the conditions under which such maps are continuous. Additionally, we prove that any unital cyclicity-preserving linear map is multiplicative. Conversely, we also characterize when a multiplicative linear map is unital cyclicity preserving. These results serve as a generalization of the Gleason-Kahane-& Zdot;elazko theorem to the setting of multiplicative maps between two RKHS. We present the composition operator as a natural class of examples of multiplicative linear maps on an RKHS. We also prove that every continuous multiplicative linear operator can be realized as a composition operator on various analytic Hilbert spaces over the unit disc $\mathbb {D}.$

On reproducing kernel Hilbert spaces with normalized complete Pick kernel, we establish an equivalent result to the Gleason–Kahane–Żelazko theorem without assuming linearity. On the way of establishing this, we observe that linearity on a multiplier algebra is enough to conclude linearity on the whole Hilbert space. By constructing a counter-example, we show that the condition of complete Pick kernel can not be removed. Also, we demonstrate the automatic continuity of such functionals. Leveraging these findings, we extend the Kowalski–Słodkowski theorem in this setup.

Let A be a complex unital Banach algebra and M be a left A-module. Let Lambda : M -> C be a map that is not necessarily linear. We establish conditions for Lambda to be linear and of multiplicative kind, from its behavior on a small subset of M. We do not assume Lambda to be continuous throughout. As an application, we give a characterization of weighted composition operators on the Hardy space H-infinity.

Let \(\mathscr {A}\) be a complex Banach algebra with unit e. Let p be a non trivial idempotent element in \(\mathscr {A}\) and \(\varepsilon >0.\) For \(a \in \mathscr {A},\) it is proved that the interior of the level set of \((p,e-p)-\varepsilon \) pseudo spectrum of a is empty in the unbounded component of \((p,e-p)\) resolvent set of a. An example is constructed to show that the condition ‘unbounded component’ can not be dropped. Further, it is proved this ‘unbounded component’ can be dropped in the case when \(\mathscr {A}\) is B(X) where X is a complex uniformly convex Banach space. That is, if \(T \in B(X)\) then interior of the level set of \((p,I-p)-\varepsilon \) pseudo spectrum is empty in \((p,I-p)\) resolvent set of T.

Let ${\mathcal{A}}$ be a complex unital Banach algebra, let $a$ be an element in it and let $0<\unicode[STIX]{x1D716}<1$ . In this article, we study the upper and lower hemicontinuity and joint continuity of the condition spectrum and its level set maps in appropriate settings. We emphasize that the empty interior of the $\unicode[STIX]{x1D716}$ -level set of a condition spectrum at a given $(\unicode[STIX]{x1D716},a)$ plays a pivotal role in the continuity of the required maps at that point. Further, uniform continuity of the condition spectrum map is obtained in the domain of normal matrices.

Conditionspectrum measures the computational stability of solving a linear system. In this paper, ten comparative results involving $$\varepsilon $$ -conditionspectrum are presented. All these theorems generalize a well known eigenvalue theorem and simultaneously compare with an appropriate pseudospectra result.

We study the spectral properties of positive absolutely minimum attaining operators defined on infinite dimensional complex Hilbert spaces and using this we derive a characterization theorem for such type of operators. We construct several examples and discuss some of the properties of this class. Also, we extend this characterization theorem for general absolutely minimum attaining operators by means of the polar decomposition theorem.

We prove that the minimum attaining property of a bounded linear operator on a Hilbert space H whose minimum modulus lies in the discrete spectrum, is stable under small compact perturbations. We also observe that given a bounded operator with strictly positive essential minimum modulus, the set of compact perturbations which fail to produce a minimum attaining operator is smaller than a nowhere dense set. In fact, it is a porous set in the ideal of all compact operators on H. Further, we try to extend these stability results to perturbations by all bounded linear operators with small norm and obtain subsequent results.

In this article, we construct a commutative unital Banach algebra, in which the property ||a(2)|| = ||a||(2) is true for the invertible elements, but cannot be extended to the whole algebra. (C) 2018 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Let A be a complex unital Banach algebra. Since the set of invertible elements is open, there is an open ball around every invertible element. In this article, we investigate the Banach algebras for which the radius given by the Neumann series is optimal.

We prove the following topological properties about L (a) 1. If = 1 then L1(a) has an empty interior unless a is a scalar multiple of the unit. 2. If 0 < < 1 then L (a) has an empty interior in the unbounded component of the resolvent set of a. Further, we show that, if the Banach space X is complex uniformly convex or X∗ is complex uniformly convex, then for any operator T ∈ B(X), L (T ) has an empty interior.

Let A be a complex commutative Banach algebra with unit 1 and delta > 0. A linear map phi: A -> C is said to be delta-almost multiplicative if|phi(ab) - phi(a)phi(b)| <= delta parallel to a parallel to parallel to b parallel to for all PH for all a, b is an element of A.Let 0 < epsilon < 1. The epsilon-condition spectrum of an element a in A is defined bysigma(epsilon)(a) := {lambda is an element of C: parallel to lambda - a parallel to parallel to(lambda - a)(-1) parallel to >= 1/epsilon}with the convention that parallel to lambda - a parallel to parallel to(lambda - a)(-1) parallel to when A a is not invertible. We prove the following results connecting these two notions:(1) If phi(1) = 1 and phi is delta-almost multiplicative, then phi(a) is an element of sigma(delta)(a) for all a in A.(2) If phi is linear and phi(a) is an element of sigma(epsilon)(a) for all a in A, then phi is delta-almost multiplicative for some delta.The first result is analogous to the Gelfand theory and the last result is analogous to the classical Gleason-Kahane-Zelazko theorem.

We define a new type of spectrum, called the is an element of-condition spectrum, of an element a in a complex unital Banach algebra A assigma(is an element of)(a) := {lambda is an element of C : lambda - a is not invertible or parallel to lambda - alpha parallel to parallel to(lambda - a)(-1)parallel to >= 1/is an element ofThis is expected to be useful in solving operator equations. We show that this is a particular case of the generalized spectrum defined by Ransford [10]. This o-condition spectrum shares some properties of the usual spectrum such as nonemptiness and compactness. But at the same time it has many properties that are different from the properties of the usual spectrum. For example, the o-condition spectrum always has only a finite number of components. Also if a is not a scalar multiple of 1 then sigma(is an element of)(a) has no isolated points. Several examples are given to illustrate the main ideas.

We prove by elementary methods the following generalization of a theorem due to Gleason, Kahane, and Żelazko. Let A be a real algebra with unit 1 such that the spectrum of every element in A is bounded and let φ : A → ℂ be a linear map such that φ (1) = 1 and ( φ ( a )) 2 + ( φ ( b )) 2 ≠ 0 for all a , b in A satisfying a b = b a and a 2 + b 2 is invertible. Then φ ( a b ) = φ ( a ) φ ( b ) for all a , b in A . Similar results are proved for real and complex algebras using Ransford′s concept of generalized spectrum. With these ideas, a sufficient condition for a linear transformation to be multiplicative is established in terms of generalized spectrum.