S. H. Kulkarni

Bio: S. H. Kulkarni is an academic researcher from Indian Institute of Technology Madras. The author has contributed to research in topics: Spectrum (functional analysis) & Commutative ring. The author has an hindex of 2, co-authored 2 publications receiving 15 citations.

Almost multiplicative functions on commutative Banach algebras

Abstract: Let A be a complex commutative Banach algebra with unit 1 and δ > 0. A linear map φ : A→ C is said to be δ-almost multiplicative if |φ(ab)− φ(a)φ(b)| ≤ δ ‖a‖ ‖b‖ for all a, b ∈ A. Let 0 < e < 1. The e-condition spectrum of an element a in A is defined by σe(a) := { λ ∈ C : ‖λ− a‖ ∥∥∥(λ− a)−1∥∥∥ ≥ 1 e } . In this note, we prove following results connecting these two notions for commutative Banach algebras. (1) If φ(1) = 1 and φ is δ-almost multiplicative, then φ(a) ∈ σδ(a) for all a in A. (2) If φ is linear and φ(a) ∈ σe(a) for all a in A, then φ is δ-almost multiplicative for some δ. The first result is analogous to the Gelfand theory and the last result is analogous to the classical Gleason-Kahane-Zelazko theorem.

Gleason-kahane-Żelazko theorem for spectrally bounded algebra

Abstract: 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

Characterization of Maximal Ideals

Abstract: The promise made at the beginning of Chapter 6 that βX is to be used to characterize the maximal ideals in C(X) and in C*(X), will be fulfilled in this chapter. The key to the description of the maximal ideals in C*(X) has already been given: C*(X) is isomorphic with C(βX), and the maximal ideals in the latter ring are in one-one correspondence with the points of βX.

Linear maps preserving pseudospectrum and condition spectrum

Abstract: We discuss properties of pseudospectrum and condition spectrum of an element in a complex unital Banach algebra and its -perturbation. Sev- eral results are proved about linear maps preserving pseudospectrum/ condition spectrum. These include the following: (1) Let A,B be complex unital Banach algebras and

Automatic continuity of almost multiplicative maps between Frechet algebras

Abstract: For Frechet algebras (A, (pn)) and (B, (qn)), a linear map T : A → B is almost multiplicative with respect to (pn) and (qn), if there exists e ≥ 0 such that qn(Tab − TaTb) ≤ epn(a)pn(b), for all n ∈ N, a, b ∈ A, and it is called weakly almost multiplicative with respect to (pn) and (qn), if there exists e ≥ 0 such that for every k ∈ N, there exists n(k) ∈ N, satisfying the inequality qk(Tab − TaTb) ≤ epn(k)(a)pn(k)(b), for all a, b ∈ A. We investigate the automatic continuity of (weakly) almost multiplicative maps between certain classes of Frechet algebras, such as Banach algebras and Frechet Q-algebras. We also obtain some results on the automatic continuity of dense range (weakly) almost multiplicative maps between Frechet algebras.

Some comparative results on eigenvalues, pseudospectra and conditionspectra

Abstract: 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.