Springer Science and Business Media LLC

For a Young function $$\phi $$ and a locally compact second countable group G,  let $$L^\phi (G)$$ denote the Orlicz space on G. In this paper, we present a necessary and sufficient condition for the topological transitivity of a sequence of cosine operators $$\{C_n\}_{n=1}^{\infty }:=\{\frac{1}{2}(T^n_{g,w}+S^n_{g,w})\}_{n=1}^{\infty }$$ , defined on $$L^{\phi }(G)$$ . We investigate the conditions for a sequence of cosine operators to be topologically mixing. Further, we go on to prove a similar result for the direct sum of a sequence of cosine operators. Finally, we give an example of topologically transitive sequence of cosine operators.

In this paper, we discuss new inequalities for accretive matrices through non-standard domains. In particular, we present several relations for $$A^r$$ and $$A\sharp _rB$$ , when A, B are accretive and $$r\in (-1,0)\cup (1,2).$$ This complements the well-established discussion of such quantities for accretive matrices when $$r\in [0,1],$$ and provides accretive versions of known results for positive matrices. Among many other results, we show that the accretive matrices A, B satisfy $$\begin{aligned} \mathfrak {R}(A\sharp _rB)\le \mathfrak {R}A\sharp _r \mathfrak {R}B, r\in (-1,0)\cup (1,2). \end{aligned}$$ This, and other results, gain their significance due to the fact that they are reversed when $$r\in (0,1).$$

We obtain several upper and lower bounds for the numerical radius of sectorial matrices. We also develop several numerical radius inequalities of the sum, product and commutator of sectorial matrices. The inequalities obtained here are sharper than the existing related inequalities for general matrices. Among many other results we prove that if A is an $$n\times n$$ complex matrix with the numerical range W(A) satisfying $$W(A)\subseteq \{re^{\pm i\theta }~:~\theta _1\le \theta \le \theta _2\},$$ where $$r>0$$ and $$\theta _1,\theta _2\in \left[ 0,\pi /2\right] ,$$ then $$\begin{aligned}{} & {} \mathrm{(i)}\quad w(A) \ge \frac{csc\gamma }{2}\Vert A\Vert + \frac{csc\gamma }{2}\left| \Vert \Im (A)\Vert -\Vert \Re (A)\Vert \right| ,\,\,\text {and}\\{} & {} \mathrm{(ii)}\quad w^2(A) \ge \frac{csc^2\gamma }{4}\Vert AA^*+A^*A\Vert + \frac{csc^2\gamma }{2}\left| \Vert \Im (A)\Vert ^2-\Vert \Re (A)\Vert ^2\right| , \end{aligned}$$ where $$\gamma =\max \{\theta _2,\pi /2-\theta _1\}$$ . We also prove that if A, B are sectorial matrices with sectorial index $$\gamma \in [0,\pi /2)$$ and they are double commuting, then $$w(AB)\le \left( 1+\sin ^2\gamma \right) w(A)w(B).$$

We investigate the integrability of Banach space valued strongly measurable functions defined on [0, 1]. In the case of functions given by $$f=\sum _{n=1}^{\infty }x_n \chi _{E_n}$$ where $$x_n\in X$$ and the sets $$E_n$$ are Lebesgue measurable and pairwise disjoint subsets of [0, 1],  there are well known characterizations for Bochner and McShane integrability of f. The absolute (resp. unconditional) convergence of $$\sum _{n=1}^{\infty }x_n m(E_n)$$ is equivalent to Bochner (resp. McShane) integrability of f. We give some conditions for scalar McShane and weakly McShane integrability of f and their relation with unconditionally converging operators. We also study the space of vector valued multipliers of strongly McShane $$(\mathcal {SM})-$$ integrable functions. We prove that if X is a commutative Banach algebra, with identity e of norm one, satisfying Radon–Nikodym property and $$M: \mathcal {SM}\rightarrow \mathcal {SM}$$ is a bounded linear operator, then there exists $$g\in L^\infty ([0,1],X)$$ such that $$M(f)= fg$$ for all $$f\in \mathcal {SM}$$ . Some results on multipliers of McShane $$({\mathcal {M}})-$$ integrable functions are also derived.

This paper specifically focuses on a specific type of q-difference equations that incorporate multiple delays. The main objective is to explore the existence of positive periodic solutions using coincidence degree theory. Notably, the equation studied in this paper has relevance to important biological growth models constructed on quantum domains. The significance of this research lies in the fact that quantum domains are not translation invariant. By investigating periodic solutions on quantum domains, the paper introduces a new perspective and makes notable advancements in the related literature, which predominantly focuses on translation invariant domains. This research contributes to a better understanding of periodic dynamics in systems governed by q-difference equations with multiple delays, particularly in the context of biological growth models on quantum domains.