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P. D. Srivastava

Publications and source records attributed to P. D. Srivastava.

18 recordsLinked to original sources
arXiv2018

Spectrum and fine spectrum of generalised lower triangular triple band matrices over the sequence space $l_p$

The spectrum of triangular band matrices defined on the sequence spaces where the entries of each band is a constant or convergent sequence is well studied. In this article, the spectrum and fine spectrum of a new generalised difference operator defined by a lower triangular triple band matrix on the sequence space $l_p (1 \leq p < \infty)$ are obtained where the bands are considered as periodic sequences. The approximate point spectrum, defect spectrum, compression spectrum and the Goldberg classification of the spectrum are also discussed. Suitable examples are given in order to supplement the results. Several special cases of our findings are discussed which confirm that our study is more general and extensive.

math.FA↗
arXiv2018

On some study of the fine spectra of generalized difference operator $Δ_{a,b}$ on $\ell_p \ (1<p<\infty)$

In this paper, we determine the spectrum, the point spectrum, the continuous spectrum and the residual spectrum of the generalized difference operator $Δ_{a,b}$ on the sequence space $\ell_p \ (1< p < \infty)$ where the real sequences $a=\{a_k\}$ and $b=\{b_k\}$ are not necessarily convergent. Hence our results generalize the work given by Akhmedov and El-Shabrawy [Math. Slovaca 65~(5) (2015) 1137--1152] for the sequence space $\ell_p (1< p <\infty)$.

math.FA↗
arXiv2018

Spectra of the lower triangular matrix $\mathbb{B}(r_1,\dots , r_l; s_1, \dots, s_{l'})$ over $c_0$

The spectra and fine spectra of the lower triangular matrix $\mathbb{B}$ $(r_1,\dots , r_l;$ $ s_1, \dots, s_{l'})$ over the sequence space $c_0$ are determined. The diagonal and sub-diagonal entries of the matrix consist of two oscillatory sequences $r=(r_{k (\text{mod} \ l)+1})$ and $s= (s_{k(\text{mod} \ l')+1})$ respectively, whereas the rest of the entries of the matrix are zero. In particular, the spectra and fine spectra of the lower triangular matrix $\mathbb{B}(r_1,\dots , r_4; s_1, \dots, s_{6})$ over $c_0$ are discussed.

math.FA↗
arXiv2018

A class of sequence spaces defined by $l$-fractional difference operator

In this paper, we generalize the fractional order difference operator using $l$- Pochhammer symbol and define $l$- fractional difference operator. The $l$- fractional difference operator is further used to introduce a class of difference sequence spaces. Some topological properties and duals of the newly defined spaces are studied.

math.FA↗
arXiv2017

On some study of the Fine Spectra of $n$-th band triangular matrices

It has been observed that for the 2nd and 3rd band lower triangular matrices $B(r,s)$ and $B(r,s,t)$, only the boundary of the spectrum gives the continuous spectrum while the rest of the entire interior region gives the residual spectrum over the sequence spaces $c_0$, $l_p$ and $bv_p$. The main focus of our present study is to investigate the possibilities of the occurrence of the similar kinds of behavior for the cases of $n (\ge4)$ band lower triangular matrices over the sequence spaces $c_0$, $l_p$ and $bv_p$. The outcomes depicts that not only the boundary part but a finite set from the interior region of the spectrum is included in the continuous spectrum while the same set is excluded from the residual spectrum. In this context, we have proved an interesting result regarding the image of the closed unit disk $|z|\le 1$ under a polynomial of degree $n\ge 1$ which plays the key role in our study. Similar studies has also been done for the sequence spaces $c$, $l_1$, $bv$ and $l_\infty$. Upper triangular matrices has also been investigated for some sequence spaces.

math.SP↗
arXiv2017

A note on Anderson's theorem in the infinite-dimensional setting

Anderson's theorem states that if the numerical range W(A) of an n-by-n matrix A is contained in the unit disk and intersects with the unit circle at more than n points, then it coincides with the (closed) unit dissk. An analogue of this result for compact A in an infinite dimensional setting was established by Gau and Wu. We consider here the case of A being the sum of a normal and compact operator.

math.FA↗
arXiv2016

On the boundary of the numerical range of some Jacobi operators

In this paper, we study the numerical range of Jacobi operators and it is shown that under certain conditions, the boundary of the numerical range of these operators can be non-round only at the points where it touches the essential spectrum. It is further shown that these points cannot be the eigenvalue of the Jacobi operators.

math.SP↗
arXiv2016

Weighted $βγ$-summability of fuzzy functions of order $θ$

The concept of weighted $βγ$ - summability of order $θ$ in case of fuzzy functions is introduced and classified into ordinary and absolute sense. Several inclusion relations among the sets are investigated. Also we have found some suitable conditions to get its relation with the generalized statistical convergence. Finally we have proved a generalized version of Tauberian theorem.

math.GM↗
arXiv2016

Some Paranormed Difference Sequence Spaces of Order $m$ Derived by Generalized Means and Compact Operators

We have introduced a new sequence space $l(r, s, t, p ;Δ^{(m)})$ combining by using generalized means and difference operator of order $m$. We have shown that the space $l(r, s, t, p ;Δ^{(m)})$ is complete under some suitable paranorm and it has Schauder basis. Furthermore, the $α$-, $β$-, $γ$- duals of this space is computed and also obtained necessary and sufficient conditions for some matrix transformations from $l(r, s, t, p; Δ^{(m)})$ to $l_{\infty}, l_1$. Finally, we obtained some identities or estimates for the operator norms and the Hausdorff measure of noncompactness of some matrix operators on the BK space $l_{p}(r, s, t ;Δ^{(m)})$ by applying the Hausdorff measure of noncompactness.

math.FA↗
arXiv2015

Certain properties of bounded variation of sequences of fuzzy numbers by using generalized weighted mean

The class of bounded variation $bv^F(u,v)$ of fuzzy numbers introduced by [8] has been investigated further with the help of the generalized weighted mean matrix $G(u,v)$. Imposing some restrictions on the matrix $G(u,v)$, we have established it's relation with different class of sequences such as our known classical sets, set of all statistically null difference sequences, Cesaro sequences etc. Also we have examined the concepts like equivalent fuzzy number, symmetric fuzzy number on this quasilinear space.

math.GM↗
arXiv2015

Some characterizations on weighted $αβ$-statistical convergence of fuzzy functions of order $θ$

Based on the concept of new type of statistical convergence defined by Aktuglu, we have introduced the weighted $αβ$ - statistical convergence of order $θ$ in case of fuzzy functions and classified it into pointwise, uniform and equi-statistical convergence. We have checked some basic properties and then the convergence are investigated in terms of their $α$-cuts. The interrelation among them are also established. We have also proved that continuity, boundedness etc are preserved in the equi-statistical sense under some suitable conditions, but not in pointwise sense.

math.GM↗
arXiv2014

On some geometric properties of generalized Musielak-Orlicz sequence space and corresponding operator ideals

Let $\boldΦ=(ϕ_n)$ be a Musielak-Orlicz function, $X$ be a real Banach space and $A$ be any infinite matrix. In this paper, a generalized vector-valued Musielak-Orlicz sequence space $l_{\bold Φ}^{A}(X)$ is introduced. It is shown that the space is complete normed linear space under certain conditions on the matrix $A$. It is also shown that $l_{\boldΦ}^{A}(X)$ is a $σ$- Dedikind complete whenever $X$ is so. We have discussed some geometric properties, namely, uniformly monotone, uniform Opial property for this space. Using the sequence of $s$-number (in the sense of Pietsch), the operators of $s$-type $l_{\boldΦ}^{A}$ and operator ideals under certain conditions on the matrix $A$ are discussed.

math.FA↗
arXiv2013

Some $B$-Difference Sequence Spaces Derived by Using Generalized Means and Compact Operators

This paper presents new sequence spaces $X(r, s, t, p ; B)$ for $X \in \{l_\infty(p), c(p), c_0(p), l(p)\}$ defined by using generalized means and difference operator. It is shown that these spaces are complete paranormed spaces and the spaces $X(r, s, t, p ; B)$ for $X \in \{c(p), c_0(p), l(p)\}$ have Schauder basis. Furthermore, the $α$-, $β$-, $γ$- duals of these sequence spaces are computed and also obtained necessary and sufficient conditions for some matrix transformations from $X(r, s, t, p ;B)$ to $X$. Finally, some classes of compact operators on the space $l_p(r, s, t ;B)$ are characterized by using the Hausdorff measure of noncompactness.

math.FA↗
arXiv2013

"Some $m$th-order Difference Sequence Spaces of Generalized Means and Compact Operators"

In this paper, new sequence spaces $X(r, s, t ;Δ^{(m)})$ for $X\in \{l_\infty, c, c_0\}$ defined by using generalized means and difference operator of order $m$ are introduced. It is shown that these spaces are complete normed linear spaces and the spaces $c_0(r, s, t ;Δ^{(m)})$, $c(r, s, t ;Δ^{(m)})$ have Schauder basis. Furthermore, the $α$-, $β$-, $γ$- duals of these spaces are computed and also obtained necessary and sufficient conditions for some matrix transformations from $X(r, s, t ;Δ^{(m)})$ to $X$. Finally, some classes of compact operators on the spaces $c_0(r, s, t ;Δ^{(m)})$ and $l_{\infty}(r, s, t ;Δ^{(m)})$ are characterized by using the Hausdorff measure of noncompactness.

math.FA↗
arXiv2013

Difference Sequence Spaces Derived by Using Generalized Means

This paper deals with new sequence spaces $X(r, s, t ;Δ) $ for $X\in \{l_\infty, c, c_0\}$ defined by using generalized means and difference operator. It is shown that these spaces are complete normed linear spaces and the spaces $X(r, s, t ;Δ)$ for $X\in \{c, c_0\}$ have Schauder basis. Furthermore, the $α$-, $β$-, $γ$- duals of these sequence spaces are computed and also established necessary and sufficient conditions for matrix transformations from $X(r, s, t ;Δ)$ to $X$.

math.FA↗
arXiv2013

Some Paranormed Difference Sequence Spaces Derived by Using Generalized Means

This paper presents new sequence spaces $X(r, s, t, p ;Δ)$ for $X \in \{l_\infty(p), c(p), c_0(p), l(p)\}$ defined by using generalized means and difference operator. It is shown that these spaces are complete under a suitable paranorm. Furthermore, the $α$-, $β$-, $γ$- duals of these sequence spaces are computed and also obtained necessary and sufficient conditions for some matrix transformations from $X(r, s, t, p ;Δ)$ to $X$. Finally, it is proved that the sequence space $l(r, s, t, p ;Δ)$ is rotund when $p_n>1$ for all $n$ and has the Kadec-Klee property.

math.FA↗