scholarly journals Functional Calculus on Real Interpolation Spaces for the Series Generators of C_0-groups

2019 ◽  
Vol 11 (5) ◽  
pp. 52
Author(s):  
Simon Joseph ◽  
Manal Juma ◽  
Isra Mukhtar ◽  
Nagat Suoliman ◽  
Fatin Saeed
Keyword(s):  
Banach Space ◽  
Functional Calculus ◽  
Real Interpolation ◽  
Interpolation Spaces ◽  
Transference Principle

In this paper, discus functional calculus properties of C_0-groups on real interpolation spaces using transference principles. Obtain interpolation versions of the classical transference principle for bounded groups and of a recent transference principle for unbounded groups. Then showed in (Markus, H., & Jan, R. 2016) that all group sequence of generators on a Banach space has a bounded H_0^∞-calculus on real interpolation spaces. Additional results are derived from this.

scholarly journals On continuity properties of semigroups in real interpolation spaces

2020 ◽  
Author(s):  
Peer Christian Kunstmann
Keyword(s):  
Banach Space ◽  
Schrodinger Equation ◽  
Continuous Semigroup ◽  
Stokes Equations ◽  
Nonlinear Schrodinger Equation ◽  
Real Interpolation ◽  
Navier Stokes ◽  
Interpolation Spaces ◽  
Navier Stokes Equations ◽  
Continuity Properties

AbstractStarting from a bi-continuous semigroup in a Banach space X (which might actually be strongly continuous), we investigate continuity properties of the semigroup that is induced in real interpolation spaces between X and the domain D(A) of the generator. Of particular interest is the case $$(X,D(A))_{\theta ,\infty }$$ ( X , D ( A ) ) θ , ∞ . We obtain topologies with respect to which the induced semigroup is bi-continuous, among them topologies induced by a variety of norms. We illustrate our results with applications to a nonlinear Schrödinger equation and to the Navier–Stokes equations on $$\mathbb {R}^d$$ R d .

scholarly journals The absolute functional calculus for sectorial operators

2005 ◽  
Author(s):  
◽  
Tamara Kucherenko
Keyword(s):  
Functional Calculus ◽  
Maximal Regularity ◽  
Real Interpolation ◽  
Future Research ◽  
Interpolation Spaces ◽  
Holomorphic Functional Calculus ◽  
Sectorial Operators ◽  
The Common ◽  
The Absolute ◽  
Key Theorem

We introduce the absolute functional calculus for sectorial operators. This notion is stronger than the common holomorphic functional calculus. We are able to improve a key theorem related to the maximal regularity problem and hence demonstrate the power and usefulness of our new concept. In trying to characterize spaces where sectorial operators have absolute calculus, we find that certain real interpolation spaces play a central role. We are then extending various known results in this setting. The idea of unifying theorems about sectorial operators on real interpolation spaces permeates our work and opens paths for future research on this subject.

scholarly journals Regular Banach space net and abstract-valued Orlicz space of range-varying type

2019 ◽  
Vol 17 (1) ◽  
pp. 1680-1702
Author(s):  
Qinghua Zhang ◽  
Yueping Zhu
Keyword(s):  
Banach Space ◽  
Orlicz Space ◽  
Orlicz Spaces ◽  
Real Interpolation ◽  
Interpolation Spaces ◽  
Type Ii ◽  
The Real ◽  
Geometrical Properties

Abstract This paper investigates the abstract-valued Orlicz space of range-varying type. We firstly give the notions and examples of partially continuous modular net and regular Banach space net of type (II), then deal with the definitions, constructions, and geometrical properties of the range-varying Orlicz spaces, including representation of the dual $\begin{array}{} L_{+}^{\varphi} \end{array}$(I, Xθ(⋅))*, and reflexivity of Lφ(I, Xθ(⋅)), under some reasonable conditions. As an application, we finally make another approach to the real interpolation spaces constructed by a generalized Φ-function.

scholarly journals Functional Calculus for the Series of Semigroup Generators Via Transference

2019 ◽  
pp. 57-84
Author(s):  
Shawgy Hussein ◽  
Simon Joseph ◽  
Ahmed Sufyan ◽  
Murtada Amin ◽  
Ranya Tahire ◽  
...  
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Half Plane ◽  
Functional Calculus ◽  
Holomorphic Functions ◽  
Transference Principle ◽  
Semigroup Generators ◽  
Bounded Holomorphic ◽  
Semigroup Generator

In this paper, apply an established transference principle to obtain the boundedness of certain functional calculi for the sequence of semigroup generators. It is proved that if be the sequence generates 0- semigroups on a Hilbert space, then for each the sequence of operators has bounded calculus for the closed ideal of bounded holomorphic functions on right half–plane. The bounded of this calculus grows at most logarithmically as. As a consequence decay at ∞. Then showed that each sequence of semigroup generator has a so-called (strong) m-bounded calculus for all m∈ℕ, and that this property characterizes the sequence of semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called semigroups, the Hilbert space results actually hold in general Banach spaces.

scholarly journals Approximation of positive operators by analytic vectors

2020 ◽  
Vol 12 (2) ◽  
pp. 412-418
Author(s):  
M.I. Dmytryshyn
Keyword(s):  
Banach Space ◽  
Positive Operator ◽  
Invariant Subspaces ◽  
Positive Operators ◽  
Interpolation Spaces ◽  
Jackson Type ◽  
Normalization Factor ◽  
Approximation Spaces ◽  
Similar Role ◽  
Approximation Errors

We give the estimates of approximation errors while approximating of a positive operator $A$ in a Banach space by analytic vectors. Our main results are formulated in the form of Bernstein and Jackson type inequalities with explicitly calculated constants. We consider the classes of invariant subspaces ${\mathcal E}_{q,p}^{\nu,\alpha}(A)$ of analytic vectors of $A$ and the special scale of approximation spaces $\mathcal {B}_{q,p,\tau}^{s,\alpha}(A)$ associated with the complex degrees of positive operator. The approximation spaces are determined by $E$-functional, that plays a similar role as the module of smoothness. We show that the approximation spaces can be considered as interpolation spaces generated by $K$-method of real interpolation. The constants in the Bernstein and Jackson type inequalities are expressed using the normalization factor.

scholarly journals Modified cauchy kernels and functional calculus for operators on Banach space

1997 ◽  
Vol 63 (1) ◽  
pp. 91-99 ◽  
Author(s):  
Edwin Franks
Keyword(s):  
Banach Space ◽  
Functional Calculus ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Cauchy Kernel ◽  
Necessary And Sufficient ◽  
Banach Space Operators

AbstractIn Banach space operators with a bounded H∞ functional calculus, Cowling et al. provide some necessary and sufficient conditions for a type-ω operator to have a bounded H∞ functional calculus. We provide an alternate development of some of their ideas using a modified Cauchy kernel which is L1 with respect to the measure ]dz]/]z]. The method is direct and has the advantage that no transforms of the functions are necessary.

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