scholarly journals Description of the structure of singular spectrum for Friedrichs model operator near singular point

2001 ◽  
Vol 28 (1) ◽  
pp. 25-40
Author(s):  
Serguei I. Iakovlev
Keyword(s):  
Analytic Function ◽  
Singular Point ◽  
Uniqueness Theorem ◽  
Analytic Functions ◽  
Point Spectrum ◽  
Positive Imaginary Part ◽  
Model Operator ◽  
Real Roots ◽  
Singular Spectrum ◽  
Friedrichs Model

The study of the point spectrum and the singular continuous one is reduced to investigating the structure of the real roots set of an analytic function with positive imaginary partM(λ). We prove a uniqueness theorem for such a class of analytic functions. Combining this theorem with a lemma on smoothness ofM(λ)near its real roots permits us to describe the density of the singular spectrum.

scholarly journals Examples of Friedrichs model operators with a cluster point of eigenvalues

2003 ◽  
Vol 2003 (10) ◽  
pp. 625-638 ◽  
Author(s):  
Serguei I. Iakovlev
Keyword(s):  
Singular Point ◽  
Modulus Of Continuity ◽  
Finiteness Condition ◽  
Cluster Point ◽  
Selfadjoint Operators ◽  
Singular Spectrum ◽  
Friedrichs Model

A family of selfadjoint operators of the Friedrichs model is considered. These symmetric type operators have one singular point, zero of orderm. For everym>3/2, we construct a rank 1 perturbation from the classLip1such that the corresponding operator has a sequence of eigenvalues converging to zero. Thus, near the singular point, there is no singular spectrum finiteness condition in terms of a modulus of continuity of a perturbation for these operators in case ofm>3/2.

scholarly journals Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator

2021 ◽  
Vol 19 (1) ◽  
pp. 329-337
Author(s):  
Huo Tang ◽  
Kaliappan Vijaya ◽  
Gangadharan Murugusundaramoorthy ◽  
Srikandan Sivasubramanian
Keyword(s):  
Analytic Function ◽  
Differential Operator ◽  
Lower Bounds ◽  
Analytic Functions ◽  
Function Class ◽  
Partial Sums ◽  
Generalized Function ◽  
Inclusion Relations ◽  
Alpha Beta

Abstract Let f k ( z ) = z + ∑ n = 2 k a n z n {f}_{k}\left(z)=z+{\sum }_{n=2}^{k}{a}_{n}{z}^{n} be the sequence of partial sums of the analytic function f ( z ) = z + ∑ n = 2 ∞ a n z n f\left(z)=z+{\sum }_{n=2}^{\infty }{a}_{n}{z}^{n} . In this paper, we determine sharp lower bounds for Re { f ( z ) / f k ( z ) } {\rm{Re}}\{f\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}\left(z)\} , Re { f k ( z ) / f ( z ) } {\rm{Re}}\{{f}_{k}\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}f\left(z)\} , Re { f ′ ( z ) / f k ′ ( z ) } {\rm{Re}}\{{f}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}^{^{\prime} }\left(z)\} and Re { f k ′ ( z ) / f ′ ( z ) } {\rm{Re}}\{{f}_{k}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}^{^{\prime} }\left(z)\} , where f ( z ) f\left(z) belongs to the subclass J p , q m ( μ , α , β ) {{\mathcal{J}}}_{p,q}^{m}\left(\mu ,\alpha ,\beta ) of analytic functions, defined by Sălăgean ( p , q ) \left(p,q) -differential operator. In addition, the inclusion relations involving N δ ( e ) {N}_{\delta }\left(e) of this generalized function class are considered.

scholarly journals Second Hankel determinant for a class of analytic functions defined by q-derivative operator

2019 ◽  
Vol 27 (2) ◽  
pp. 167-177
Author(s):  
Dorina Răducanu
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Hankel Determinant ◽  
Derivative Operator ◽  
Second Hankel Determinant

AbstractIn this paper, we obtain the estimates for the second Hankel determinant for a class of analytic functions defined by q-derivative operator and subordinate to an analytic function.

LIPSCHITZ-TYPE INEQUALITIES FOR ANALYTIC FUNCTIONS IN BANACH ALGEBRAS

2019 ◽  
Vol 100 (3) ◽  
pp. 489-497
Author(s):  
SILVESTRU SEVER DRAGOMIR
Keyword(s):  
Analytic Function ◽  
Banach Algebra ◽  
Analytic Functions ◽  
Banach Algebras ◽  
Logarithmic Functions

In this paper we provide some bounds for the quantity $\Vert f(y)-f(x)\Vert$, where $f:D\rightarrow \mathbb{C}$ is an analytic function on the domain $D\subset \mathbb{C}$ and $x$, $y\in {\mathcal{B}}$, a Banach algebra, with the spectra $\unicode[STIX]{x1D70E}(x)$, $\unicode[STIX]{x1D70E}(y)\subset D$. Applications for the exponential and logarithmic functions on the Banach algebra ${\mathcal{B}}$ are also given.

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