scholarly journals Resolvent for Non-Self-Adjoint Differential Operator with Block-Triangular Operator Potential

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Aleksandr Mikhailovich Kholkin
Keyword(s):  
Differential Operator ◽  
Sufficient Conditions ◽  
Operator Potential ◽  
Triangular Operator

A resolvent for a non-self-adjoint differential operator with a block-triangular operator potential, increasing at infinity, is constructed. Sufficient conditions under which the spectrum is real and discrete are obtained.

scholarly journals Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular Operator Coefficients

2021 ◽  
Author(s):  
Aleksandr Kholkin
Keyword(s):  
Asymptotic Behavior ◽  
Differential Operator ◽  
Liouville Equation ◽  
Spectral Properties ◽  
Sufficient Conditions ◽  
Triangular Matrix ◽  
Spectral Singularities ◽  
Matrix Potential ◽  
Triangular Operator ◽  
Sturm Liouville

In this chapter, the Sturm-Liouville equation with block-triangular, increasing at infinity operator potential is considered. A fundamental system of solutions is constructed, one of which decreases at infinity, and the second increases. The asymptotic behavior at infinity was found out. The Green’s function and the resolvent for a non-self-adjoint differential operator are constructed. This allows to obtain sufficient conditions under which the spectrum of this non-self-adjoint differential operator is real and discrete. For a non-self-adjoint Sturm-Liouville operator with a triangular matrix potential growing at infinity, an example of operator having spectral singularities is constructed.

scholarly journals Property $(w)$ of upper triangular operator Matrices

2020 ◽  
Vol 51 (2) ◽  
pp. 81-99
Author(s):  
Mohammad M.H Rashid
Keyword(s):  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Extension Property ◽  
Operator Matrices ◽  
Single Valued Extension Property ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Number Of Authors

Let $M_C=\begin{pmatrix} A & C \\ 0 & B \\ \end{pmatrix}\in\LB(\x,\y)$ be be an upper triangulate Banach spaceoperator. The relationship between the spectra of $M_C$ and $M_0,$ and theirvarious distinguished parts, has been studied by a large number of authors inthe recent past. This paper brings forth the important role played by SVEP,the {\it single-valued extension property,} in the study of some of these relations. In this work, we prove necessary and sufficient conditions of implication of the type $M_0$ satisfies property $(w)$ $\Leftrightarrow$ $M_C$ satisfies property $(w)$ to hold. Moreover, we explore certain conditions on $T\in\LB(\hh)$ and $S\in\LB(\K)$ so that the direct sum $T\oplus S$ obeys property $(w)$, where $\hh$ and $\K$ are Hilbert spaces.

scholarly journals Some Janowski Type Harmonic q-Starlike Functions Associated with Symmetrical Points

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 629 ◽  
Author(s):  
Muhammad Arif ◽  
Omar Barkub ◽  
Hari Srivastava ◽  
Saleem Abdullah ◽  
Sher Khan
Keyword(s):  
Differential Operator ◽  
Harmonic Functions ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Partial Sums ◽  
Circular Region ◽  
Necessary And Sufficient ◽  
New Criterion ◽  
Symmetrical Points ◽  
Convexity Conditions

The motive behind this article is to apply the notions of q-derivative by introducing some new families of harmonic functions associated with the symmetric circular region. We develop a new criterion for sense preserving and hence the univalency in terms of q-differential operator. The necessary and sufficient conditions are established for univalency for this newly defined class. We also discuss some other interesting properties such as distortion limits, convolution preserving, and convexity conditions. Further, by using sufficient inequality, we establish sharp bounds of the real parts of the ratios of harmonic functions to its sequences of partial sums. Some known consequences of the main results are also obtained by varying the parameters.

Necessary and sufficient conditions for asymptotic decay of oscillations in delayed functional equations

1981 ◽  
Vol 91 (1-2) ◽  
pp. 135-145
Author(s):  
Lu-San Chen ◽  
Cheh-Chih Yeh
Keyword(s):  
Differential Operator ◽  
Functional Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Oscillatory Solutions ◽  
Asymptotic Decay ◽  
Image Position ◽  
Necessary And Sufficient

SynopsisThis paper studies the equationwhere the differential operator Ln is defined byand a necessary and sufficient condition that all oscillatory solutions of the above equation converge to zero asymptotically is presented. The results obtained extend and improve previous ones of Kusano and Onose, and Singh, even in the usual case wherewhere N is an integer with l≦N≦n–1.

scholarly journals Intrinsic compound kernel estimates for the transition probability density of Lévy-type processes and their applications

2018 ◽  
Vol 37 (1) ◽  
pp. 53-100 ◽  
Author(s):  
Victoria Knopova ◽  
Aleksei Kulik
Keyword(s):  
Differential Operator ◽  
Probability Density ◽  
Borel Measure ◽  
Transition Probability ◽  
Sufficient Conditions ◽  
Feller Process ◽  
Kernel Estimates ◽  
Transition Probability Density

INTRINSIC COMPOUND KERNEL ESTIMATES FOR THE TRANSITION PROBABILITY DENSITY OF LÉVY-TYPE PROCESSES AND THEIR APPLICATIONSStarting with an integro-differential operator L, C2∞ ℜn, we prove that its C∞ℜn-closure is the generator of a Feller process X, which admits a transition probability density. To construct this transition probability density, we develop a version of the parametrix method and a verification procedure, which proves that the constructed object is the claimed one. As a part of the construction, we prove the intrinsic upper and lower estimates on the density. As an application of the constructed estimates we state the necessary and separately sufficient conditions under which a given Borel measure belongs to the Kato and Dynkin classes with respect to the constructed transition probability density.

scholarly journals Riesz Inequality for the System of Root Functions of Second Order Ordinary Differential Operator

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Differential Operator ◽  
Sufficient Conditions ◽  
Second Order ◽  
Ordinary Differential Operator ◽  
Root Functions ◽  
Riesz Property ◽  
System Of Root Functions ◽  
The Given ◽  
Riesz Inequality

An ordinary differential operator of second order with coefficients is considered. The Riesz property of the system of root functions of the given operator is studied. The criterion of Bessel property in 2 L , of root functions system is established and use it to obtain sufficient conditions for the Riesz property of a system of normalized root functions of this operator in p L .

25.—Spectrum of a Third-order Differential Operator with Large Coefficients

1975 ◽  
Vol 72 (4) ◽  
pp. 299-305
Author(s):  
K. Unsworth
Keyword(s):  
Differential Operator ◽  
Differential Expression ◽  
Real Axis ◽  
Sufficient Conditions ◽  
Order Differential Operator ◽  
Third Order ◽  
Minimal Operator ◽  
The Third ◽  
Entire Real Axis

SynopsisThis paper sets out to study the spectrum of self-adjoint extensions of the minimal operator associated with the third-order formally symmetric differential expression. The technique employed is the method of singular sequences. Sufficient conditions are established on the coefficients of the differential expression in order that the spectrum should cover the entire real axis. Particular cases in which the coefficients behave roughly as powers of x as the magnitude of x becomes large are then considered, and certain conclusions are drawn regarding the spectra under different restrictions on these powers of x.

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