Poisson like matrix operator and its application in p-summable space

Abstract

Abstract The incomplete gamma function Γ(a, u) is defined by Γ ( a , u ) = ∫ u ∞ t a − 1 e − t d t , $$\Gamma(a,u)=\int\limits_{u}^{\infty}t^{a-1}\textrm{e}^{-t}\textrm{d} t,$$ where u > 0. Using the incomplete gamma function, we define a new Poisson like regular matrix P ( μ ) = ( p n k μ ) $\mathfrak{P}(\mu)=(p^{\mu}_{nk})$ given by p n k μ = n ! Γ ( n + 1 , μ ) e − μ μ k k ! ( 0 ≤ k ≤ n ) , 0 ( k > n ) , $$p^{\mu}_{nk}= \begin{cases} \dfrac{n!}{\Gamma(n+1,\mu)}\dfrac{\textrm{e}^{-\mu}\mu^k}{k!} \quad &(0\leq k\leq n), \\[1ex] 0\quad & (k>n), \end{cases}$$ where μ > 0 is fixed. We introduce the sequence space ℓ p ( P ( μ ) ) $\ell_p(\mathfrak{P}(\mu))$ for 1 ≤ p ≤ ∞ and some topological properties, inclusion relations and generalized duals of the newly defined space are discussed. Also we characterize certain matrix classes and compact operators related to the space ℓ p ( P ( μ ) ) $\ell_p(\mathfrak{P}(\mu))$ . We obtain Gurarii’s modulus of convexity and investigate some geometric properties of the new space. Finally, spectrum of the operator P ( μ ) $\mathfrak{P}(\mu)$ on sequence space c 0 has been investigated.