On the Morse–Bott property of analytic functions on Banach spaces with Łojasiewicz exponent one half

Let f : (ℝn, 0) → (ℝ, 0) be a nonconstant analytic function defined in a neighborhood of the origin 0 ∈ ℝn. The classical Łojasiewicz inequality states that there exist positive constants δ, c and l such that |f(x)| ≥ cd(x, f-1(0))l for ‖x‖ ≤ δ, where d(x, f-1(0)) denotes the distance from x to the set f-1(0). The Łojasiewicz exponent of f at the origin 0 ∈ ℝn, denoted by [Formula: see text], is the infimum of the exponents l satisfying the Łojasiewicz inequality. In this paper, we establish a formula for computing the Łojasiewicz exponent [Formula: see text] of f in terms of the Newton polyhedron of f in the case where f is nonnegative and nondegenerate.

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Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

Axioms ◽

10.3390/axioms10030150 ◽

2021 ◽

Vol 10 (3) ◽

pp. 150

Author(s):

Andriy Zagorodnyuk ◽

Anna Hihliuk

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Analytic Functions ◽

Entire Functions ◽

Dimensional Algebra ◽

Dimensional Banach Space ◽

Infinite Dimensional ◽

Linear Subspaces ◽

Infinite Dimensional Banach Space

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

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Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions

Canadian Mathematical Bulletin ◽

10.4153/cmb-1999-016-x ◽

1999 ◽

Vol 42 (2) ◽

pp. 139-148 ◽

Cited By ~ 59

Author(s):

José Bonet ◽

Paweł Dománski ◽

Mikael Lindström

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Analytic Functions ◽

Composition Operators ◽

Essential Norm ◽

Weak Compactness ◽

Weakly Compact ◽

Point Evaluation ◽

Weighted Banach Spaces ◽

Spaces Of Analytic Functions

AbstractEvery weakly compact composition operator between weighted Banach spaces of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces.

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