scholarly journals On power-bounded prespectral operators

1976 ◽  
Vol 20 (2) ◽  
pp. 173-175
Author(s):  
H. R. Dowson
Keyword(s):  
Growth Condition ◽  
Bounded Operator ◽  
Spectral Operator ◽  
Scalar Type ◽  
Image Position ◽  
Power Bounded Operator ◽  
Power Bounded

Foguel (8) and Fixman (7) independently proved that an invertible spectral operator, which is power-bounded, is of scalar type. Their proofs rely heavily on a result of Dunford on spectral operators whose resolvents satisfy a growth condition. (See Lemma 3.16 of (6, p. 609).) Observe that the resolvent of an invertible power-bounded operator T satisfies an inequality of the form

scholarly journals THE SPECTRAL TYPE OF SUMS OF OPERATORS ON NON-HILBERTIAN BANACH LATTICES

2008 ◽  
Vol 84 (2) ◽  
pp. 193-198
Author(s):  
IAN DOUST ◽  
GILLES LANCIEN
Keyword(s):  
Hilbert Space ◽  
Large Class ◽  
Spectral Type ◽  
Bounded Operator ◽  
Banach Lattices ◽  
Spectral Operator ◽  
Real Scalar ◽  
Scalar Type

AbstractT. A. Gillespie showed that on a Hilbert space the sum of a well-bounded operator and a commuting real scalar-type spectral operator is well-bounded. A longstanding question asked whether this might still hold true for operators on Lp spaces for $1\less p\less \infty $. We show here that this conjecture is false. Indeed for a large class of reflexive spaces, the above property characterizes Hilbert space.

scholarly journals Boundedly Spaced Subsequences and Weak Dynamics

2018 ◽  
Vol 2018 ◽  
pp. 1-5
Author(s):  
C. S. Kubrusly ◽  
P. C. M. Vieira
Keyword(s):  
Hilbert Space ◽  
Unitary Operator ◽  
Bounded Operator ◽  
Weak Stability ◽  
Weakly Stable ◽  
Power Bounded Operator ◽  
Power Bounded

Weak supercyclicity is related to weak stability, which leads to the question that asks whether every weakly supercyclic power bounded operator is weakly stable. This is approached here by investigating weak l-sequential supercyclicity for Hilbert-space contractions through Nagy–Foliaş–Langer decomposition, thus reducing the problem to the quest of conditions for a weakly l-sequentially supercyclic unitary operator to be weakly stable, and this is done in light of boundedly spaced subsequences.

scholarly journals Mean ergodic operators and reflexive Fréchet lattices

2011 ◽  
Vol 141 (5) ◽  
pp. 897-920 ◽  
Author(s):  
José Bonet ◽  
Ben de Pagter ◽  
Werner J. Ricker
Keyword(s):  
Bounded Operator ◽  
Banach Lattices ◽  
Positive Power ◽  
Bounded Linear ◽  
Mean Ergodic Operators ◽  
Power Bounded Operator ◽  
Complemented Lattice ◽  
Locally Convex ◽  
Strong Dual ◽  
Power Bounded

Connections between (positive) mean ergodic operators acting in Banach lattices and properties of the underlying lattice itself are well understood (see the works of Emel'yanov, Wolff and Zaharopol). For Fréchet lattices (or more general locally convex solid Riesz spaces) there is virtually no information available. For a Fréchet lattice E, it is shown here that (amongst other things) every power-bounded linear operator on E is mean ergodic if and only if E is reflexive if and only if E is Dedekind σ-complete and every positive power-bounded operator on E is mean ergodic if and only if every positive power-bounded operator in the strong dual E′β (no longer a Fréchet lattice) is mean ergodic. An important technique is to develop criteria that detect when E admits a (positively) complemented lattice copy of c0, l1 or l∞.

Growth conditions on subharmonic functions and resolvents of operators

1985 ◽  
Vol 97 (2) ◽  
pp. 321-324
Author(s):  
J. R. Partington
Keyword(s):  
Banach Space ◽  
Numerical Range ◽  
Bounded Operator ◽  
Complex Banach Space ◽  
Growth Conditions ◽  
Subharmonic Functions ◽  
Image Position

Let X be a complex Banach space and T a bounded operator on X. The numerical range of T is defined by

scholarly journals Partial regularity for minimizers of discontinuous quasiconvex integrals with general growth

2021 ◽  
pp. 1-42
Author(s):  
Christopher Goodrich ◽  
Giovanni Scilla ◽  
Bianca Stroffolini
Keyword(s):  
Growth Condition ◽  
Partial Regularity ◽  
Hölder Continuity ◽  
Holder Continuity ◽  
General Growth ◽  
Image Position

We prove the partial Hölder continuity for minimizers of quasiconvex functionals \begin{equation*} \mathcal{F}({\bf u}) \colon =\int_{\Omega} f(x,{\bf u},D{\bf u})\,\textrm{d} x, \end{equation*} where $f$ satisfies a uniform VMO condition with respect to the $x$ -variable and is continuous with respect to ${\bf u}$ . The growth condition with respect to the gradient variable is assumed a general one.

scholarly journals On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis

2019 ◽  
Vol 17 (1) ◽  
pp. 1082-1112
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Evolution Equation ◽  
Weak Solutions ◽  
A Priori ◽  
Complex Banach Space ◽  
Spectral Operator ◽  
Scalar Type ◽  
Improvement Effect ◽  
Abstract Evolution Equation ◽  
Necessary And Sufficient

Abstract Given the abstract evolution equation $$\begin{array}{} \displaystyle y'(t)=Ay(t),\, t\ge 0, \end{array}$$ with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order β ≥ 1, in particular analytic or entire, on the open semi-axis (0, ∞). Also, revealed is a certain interesting inherent smoothness improvement effect.

scholarly journals On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator on the Real Axis

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Marat V. Markin
Keyword(s):  
Evolution Equation ◽  
Weak Solutions ◽  
Normal Operator ◽  
A Priori ◽  
Complex Banach Space ◽  
Complex Hilbert Space ◽  
Spectral Operator ◽  
Scalar Type ◽  
Improvement Effect ◽  
Abstract Evolution Equation

Given the abstract evolution equation y′(t)=Ay(t),  t∈R, with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly infinite differentiable on R. The important case of the equation with a normal operator A in a complex Hilbert space is obtained immediately as a particular case. Also, proved is the following inherent smoothness improvement effect explaining why the case of the strong finite differentiability of the weak solutions is superfluous: if every weak solution of the equation is strongly differentiable at 0, then all of them are strongly infinite differentiable on R.

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