The boundedness and compactness of a product-type operator from Zygmund-type spaces to Bloch-Orlicz spaces are investigated in this paper.

In this paper, we study if T is an (m,C)-isometric operator and CT+C commutes with T, then T+ is an (m,C)-isometric operator. We also give local spectral properties and spectral relations of (m;C)-isometric operators, such as property (?), decomposability, the single-valued extension property and Dunford?s boundedness. We also investigate perturbation of (m,C)-isometric operators by nilpotent operators and by algebraic operators and give some properties.

The exact value of the generalized von Neumann–Jordan constant is found for the Banaś–Frączek space $\mathbb {R}^2_\lambda $: $C^{(p)}_{\rm NJ}(\mathbb {R}^2_\lambda )=1+(1-{1/\lambda ^{2}})^{{p/2}}$ ($\lambda \gt 1$, $p\geq 2$, $\lambda ^2(1-{1/\lambda ^