CONTINUITY OF A CONDITION SPECTRUM AND ITS LEVEL SETS

2019 ◽  
Vol 108 (3) ◽  
pp. 412-430
Author(s):  
D. SUKUMAR ◽  
S. VEERAMANI
Keyword(s):  
Banach Algebra ◽  
Level Set ◽  
Level Sets ◽  
Uniform Continuity ◽  
Pivotal Role ◽  
Empty Interior ◽  
Joint Continuity ◽  
Normal Matrices ◽  
Unital Banach Algebra

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.

The numerical range in the second dual of a Banach algebra

1981 ◽  
Vol 89 (2) ◽  
pp. 301-307
Author(s):  
R. R. Smith
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Numerical Range ◽  
Special Property ◽  
Empty Interior ◽  
Banach Theorem ◽  
Second Dual ◽  
Algebra Theory ◽  
Unital Banach Algebra ◽  
Selection Of

An elementary consequence of the Hahn-Banach theorem is that every Banach space Y is ω*-dense in its second dual Y**, so that every element y ∈ Y** is the w*-limit of a net {ya}α ∈ Λ from Y. There is, of course, a great deal of choice in the selection of such a net, and so one may impose extra conditions on the net related to some special property of the limit point, and then ask for existence. The object of this paper is to present such a result in the context of a unital Banach algebra and its second dual , and then to give two applications to Banach algebra theory. The theorem to be proved is this: if the numerical range W(a) of an element in has non-empty interior then a is the ω*-limit of a net {aa}α ∈ Λ from whose numerical ranges are contained in W(a), while if W(a) has empty interior then the numerical ranges W(aα) are contained in a shrinking set of neighbourhoods of W(a).

A characterization of Jordan and 5-Jordan homomorphisms between Banach algebras

2018 ◽  
Vol 11 (02) ◽  
pp. 1850021 ◽  
Author(s):  
A. Zivari-Kazempour
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Jordan Homomorphism ◽  
Jordan Homomorphisms ◽  
Unital Banach Algebra

We prove that each surjective Jordan homomorphism from a Banach algebra [Formula: see text] onto a semiprime commutative Banach algebra [Formula: see text] is a homomorphism, and each 5-Jordan homomorphism from a unital Banach algebra [Formula: see text] into a semisimple commutative Banach algebra [Formula: see text] is a 5-homomorphism.

scholarly journals Bipolar Fuzzy Relations

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1044 ◽  
Author(s):  
Jeong-Gon Lee ◽  
Kul Hur
Keyword(s):  
Equivalence Class ◽  
Level Set ◽  
Equivalence Relation ◽  
Level Sets ◽  
Equivalence Classes ◽  
Fuzzy Relation ◽  
Fuzzy Relations ◽  
Transitive Relation ◽  
Fuzzy Partition

We introduce the concepts of a bipolar fuzzy reflexive, symmetric, and transitive relation. We study bipolar fuzzy analogues of many results concerning relationships between ordinary reflexive, symmetric, and transitive relations. Next, we define the concepts of a bipolar fuzzy equivalence class and a bipolar fuzzy partition, and we prove that the set of all bipolar fuzzy equivalence classes is a bipolar fuzzy partition and that the bipolar fuzzy equivalence relation is induced by a bipolar fuzzy partition. Finally, we define an ( a , b ) -level set of a bipolar fuzzy relation and investigate some relationships between bipolar fuzzy relations and their ( a , b ) -level sets.

scholarly journals On the Correlation Between Nodal and Nonzero Level Sets for Random Spherical Harmonics

2020 ◽  
Author(s):  
Domenico Marinucci ◽  
Maurizia Rossi
Keyword(s):  
Level Set ◽  
Spherical Harmonics ◽  
Level Sets ◽  
Partial Correlation

AbstractWe study the correlation between the nodal length of random spherical harmonics and the length of a nonzero level set. We show that the correlation is asymptotically zero, while the partial correlation after removing the effect of the random $$L^2$$ L 2 -norm of the eigenfunctions is asymptotically one.

scholarly journals The Lyapunov spectrum of some parabolic systems

2009 ◽  
Vol 29 (3) ◽  
pp. 919-940 ◽  
Author(s):  
KATRIN GELFERT ◽  
MICHAŁ RAMS
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Level Set ◽  
Level Sets ◽  
Parabolic Systems ◽  
Lyapunov Spectrum ◽  
Interval Maps ◽  
Set Of Points

AbstractWe study the Hausdorff dimension for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topological entropy.

On Banach algebra elements of thin numerical range

1979 ◽  
Vol 86 (1) ◽  
pp. 71-83 ◽  
Author(s):  
R. R. Smith
Keyword(s):  
Banach Algebra ◽  
Numerical Range ◽  
Natural Generalization ◽  
The Real ◽  
C Algebra ◽  
Image Position ◽  
Unital Banach Algebra ◽  
Real Subspace

Among the elements of a complex unital Banach algebra the real subspace of hermitian elements deserves special attention. This forms the natural generalization of the set of self-adjoint elements in a C*-algebra and exhibits many of the same properties. Two equivalent definitions may be given: if W(h) ⊂ , where W(h) denotes the numerical range of h (7), or if ║eiλh║ = 1 for all λ ∈ . In this paper some related subsets are introduced and studied. For δ ≥ 0, an element is said to be a member of if the conditionis satisfied. These may be termed the elements of thin numerical range if δ is small.

scholarly journals Polynomials in a hermitian element

1988 ◽  
Vol 30 (2) ◽  
pp. 171-176 ◽  
Author(s):  
M. J. Crabb ◽  
C. M. McGregor
Keyword(s):  
Banach Algebra ◽  
Dual Space ◽  
Numerical Range ◽  
Numerical Radius ◽  
Hermitian Element ◽  
Unital Banach Algebra

For an element a of a unital Banach algebra A with dual space A′, we define the numerical range V(a) = {f(a):f ∊ A′, ∥f∥ = f(1) = 1}, and the numerical radius v(a) = sup{⃒z⃒:z ∊ V(a)}. An element a is said to be Hermitian if V(a) ⊆ ℝ ,equivalently ∥exp (ita)∥ = 1(t ∊ ℝ). Under the condition V(h) ⊆ [-1, 1], any polynomial in h attains its greatest norm in the algebra Ea[-1,1], generated by an element h with V(h) = [-1, 1].

scholarly journals The Geometry of the Weak Lefschetz Property and Level Sets of Points

2008 ◽  
Vol 60 (2) ◽  
pp. 391-411 ◽  
Author(s):  
Juan C. Migliore
Keyword(s):  
Level Set ◽  
Level Sets ◽  
Hilbert Function ◽  
Worst Case ◽  
Weak Lefschetz Property ◽  
Lefschetz Property ◽  
Strong Lefschetz Property ◽  
Set Of Points

AbstractIn a recent paper, F. Zanello showed that level Artinian algebras in 3 variables can fail to have the Weak Lefschetz Property (WLP), and can even fail to have unimodal Hilbert function. We show that the same is true for the Artinian reduction of reduced, level sets of points in projective 3-space. Our main goal is to begin an understanding of how the geometry of a set of points can prevent its Artinian reduction from having WLP, which in itself is a very algebraic notion. More precisely, we produce level sets of points whose Artinian reductions have socle types 3 and 4 and arbitrary socle degree ≥ 12 (in the worst case), but fail to have WLP. We also produce a level set of points whose Artinian reduction fails to have unimodal Hilbert function; our example is based on Zanello's example. Finally, we show that a level set of points can have Artinian reduction that has WLP but fails to have the Strong Lefschetz Property. While our constructions are all based on basic double G-linkage, the implementations use very different methods.

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