scholarly journals Complex powers of operators

2008 ◽  
Vol 83 (97) ◽  
pp. 15-25
Author(s):  
Marko Kostic
Keyword(s):  
Complex Plane ◽  
Analytic Semigroup ◽  
Fractional Power ◽  
Growth Order ◽  
Fractional Number ◽  
Densely Defined Operator ◽  
Complex Powers ◽  
Densely Defined

We define the complex powers of a densely defined operator A whose resolvent exists in a suitable region of the complex plane. Generally, this region is strictly contained in an angle and there exists ? ? [0,?) such that the resolvent of A is bounded by O((1 + |?|)?) there. We prove that for some particular choices of a fractional number b, the negative of the fractional power (-A)b is the c.i.g. of an analytic semigroup of growth order r > 0.

scholarly journals Complex powers of nondensely defined operators

2011 ◽  
Vol 90 (104) ◽  
pp. 47-64 ◽  
Author(s):  
Marko Kostic
Keyword(s):  
Mild Solutions ◽  
Acute Angle ◽  
Analytic Semigroups ◽  
Cauchy Problems ◽  
Growth Order ◽  
Fractional Powers ◽  
Complex Powers ◽  
Cosine Functions ◽  
Abstract Cauchy Problems ◽  
Densely Defined

The power (?A)b, b ? C is defined for a closed linear operator A whose resolvent is polynomially bounded on the region which is, in general, strictly contained in an acute angle. It is proved that all structural properties of complex powers of densely defined operators with polynomially bounded resolvent remain true in the newly arisen situation. The fractional powers are considered as generators of analytic semigroups of growth order r > 0 and applied in the study of corresponding incomplete abstract Cauchy problems. In the last section, the constructed powers are incorporated in the analysis of the existence and growth of mild solutions of operators generating fractionally integrated semigroups and cosine functions.

scholarly journals Existence of Mild Solutions for a Class of Fractional Evolution Equations with Compact Analytic Semigroup

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
He Yang
Keyword(s):  
Analytic Semigroup ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Fractional Power ◽  
Growth Conditions ◽  
Local Growth ◽  
Fractional Evolution Equations ◽  
Nonlinear Part ◽  
Compact Analytic Semigroup

This paper deals with the existence of mild solutions for a class of fractional evolution equations with compact analytic semigroup. We prove the existence of mild solutions, assuming that the nonlinear part satisfies some local growth conditions in fractional power spaces. An example is also given to illustrate the applicability of abstract results.

scholarly journals Fractional powers of operators defined on a Fréchet space

1984 ◽  
Vol 27 (2) ◽  
pp. 165-180 ◽  
Author(s):  
W. Lamb
Keyword(s):  
Banach Space ◽  
Fractional Power ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fractional Powers ◽  
Fractional Powers Of Operators ◽  
Image Position ◽  
Densely Defined

The problem of finding a suitable representation for a fractional power of an operator defined in a Banach space X has, in recent years, attracted much attention. In particular, Balakrishnan [1], Hovel and Westphal [3] and Komatsu [4] have examined the problem of defining the fractionalpower (–A)α for closed densely-defined operators A such that

scholarly journals Dilatations of Linear Operators

2020 ◽  
Vol 66 (2) ◽  
pp. 209-220
Author(s):  
Yu. L. Kudryashov
Keyword(s):  
Linear Operator ◽  
Bounded Operator ◽  
Regular Point ◽  
Explicit Construction ◽  
Linear Operators ◽  
The Self ◽  
Dissipative Operator ◽  
Bounded Linear ◽  
Densely Defined Operator ◽  
Densely Defined

The article is devoted to building various dilatations of linear operators. The explicit construction of a unitary dilation of a compression operator is considered. Then the J -unitary dilatation of a bounded operator is constructed by means of the operator knot concept of a bounded linear operator. Using the Pavlov method, we construct the self-adjoint dilatation of a bounded dissipative operator. We consider spectral and translational representations of the self-adjoint dilatation of a densely defined dissipative operator with nonempty set of regular points. Using the concept of an operator knot for a bounded operator and the Cayley transform, we introduce an operator knot for a linear operator. By means of this concept, we construct the J -self-adjoint dilatation of a densely defined operator with a regular point. We obtain conditions of isomorphism of extraneous dilations and their minimality.

scholarly journals Resolvent Positive Operators and Positive Fractional Resolvent Families

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Nan-Ding Li ◽  
Ru Liu ◽  
Miao Li
Keyword(s):  
Banach Space ◽  
Fractional Differential Equations ◽  
Ordered Banach Space ◽  
Resolvent Family ◽  
Completely Monotonic ◽  
Densely Defined Operator ◽  
Fractional Resolvent Families ◽  
Fractional Differential ◽  
Densely Defined ◽  
Resolvent Families

This paper is concerned with positive α -times resolvent families on an ordered Banach space E (with normal and generating cone), where 0 < α ≤ 2 . We show that a closed and densely defined operator A on E generates a positive exponentially bounded α -times resolvent family for some 0 < α < 1 if and only if, for some ω ∈ ℝ , when λ > ω , λ ∈ ρ A , R λ , A ≥ 0 and sup λ R λ , A : λ ≥ ω < ∞ . Moreover, we obtain that when 0 < α < 1 , a positive exponentially bounded α -times resolvent family is always analytic. While A generates a positive α -times resolvent family for some 1 < α ≤ 2 if and only if the operator λ α − 1 λ α − A − 1 is completely monotonic. By using such characterizations of positivity, we investigate the positivity-preserving of positive fractional resolvent family under positive perturbations. Some examples of positive solutions to fractional differential equations are presented to illustrate our results.

A unified treatment for Tikhonov regularization using a general stabilizing operator

2015 ◽  
Vol 13 (02) ◽  
pp. 201-215
Author(s):  
M. T. Nair
Keyword(s):  
Tikhonov Regularization ◽  
Regularization Parameter ◽  
Optimal Error Estimates ◽  
Bounded Linear ◽  
Densely Defined Operator ◽  
Special Cases ◽  
Ill Posed ◽  
Residual Norm ◽  
Well Posed ◽  
Densely Defined

While dealing with the problem of solving an ill-posed operator equation Tx = y, where T : X → Y is a bounded linear operator between Hilbert spaces X and Y, one looks for a stable method for approximating [Formula: see text], a least-residual norm solution which minimizes a seminorm x ↦ ‖Lx‖, where L : D(L) ⊆ X → X is a (possibly unbounded) closed densely defined operator in X. If the operators T and L satisfy a completion condition ‖Tx‖2 + ‖Lx‖2 ≥ γ‖x‖2 for all x ∈ D(L*L) for some constant γ > 0, then Tikhonov regularization is one of the simple and widely used of such procedures in which the regularized solution is obtained by solving a well-posed equation [Formula: see text] where yδ is a noisy data and α > 0 is the regularization parameter to be chosen appropriately. We prescribe a condition on (T, L) which unifies the analysis for ordinary Tikhonov regularization, that is, L = I, and also the case of L = Bs with B being a strictly positive closed densely defined unbounded operator which generates a Hilbert scale {Xt}t>0. Under the new framework, we provide estimates for the best possible worst error and order optimal error estimates for the regularized solutions under certain general source condition which incorporates in its fold many existing results as special cases, by choosing regularization parameter using a Morozov-type discrepancy principle.

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