Springer Science and Business Media LLC

Characterizations of the star, minus and diamond orders of operators are given in various contexts and the relationship between these orders is made more transparent. Moreover, we introduce a new partial order of operators which provides a unified scenario for studying the other three orders.

A linear mapping $$\delta $$ from a $${*}$$ -algebra $$\mathcal {A}$$ into a $${*}$$ - $$\mathcal {A}$$ -bimodule $$\mathcal {M}$$ is a $${*}$$ -derivable mapping at $$G\in \mathcal {A}$$ if $$A\delta (B)^{*}+\delta (A)B=\delta (G)$$ for each A, B in $$\mathcal {A}$$ with $$AB^{*}=G$$ . We prove that every (continuous) $${*}$$ -derivable mapping at G from a (unital $$C^{*}$$ -algebra) factor von Neumann algebra into its Banach $${*}$$ -bimodule is a $${*}$$ -derivation if and only if G is a left separating point. A linear mapping $$\delta $$ from a $${*}$$ -algebra $$\mathcal {A}$$ into a $${*}$$ -left $$\mathcal {A}$$ -module $$\mathcal {M}$$ is a $${*}$$ -left derivable mapping at $$G\in \mathcal {A}$$ if $$A\delta (B)^{*}+B\delta (A)=\delta (G)$$ for each A, B in $$\mathcal {A}$$ with $$AB^{*}=G$$ . We prove that every continuous $${*}$$ -left derivable mapping at a left separating point from a unital $$C^{*}$$ -algebra or von Neumann algebra into its Banach $${*}$$ -left $$\mathcal {A}$$ -module is identical with zero under certain conditions.

We obtain bounds for the numerical radius of $$2 \times 2$$ operator matrices which improve on the existing bounds. We also show that the inequalities obtained here generalize the existing ones. As an application of the results obtained here, we estimate the bounds for the zeros of a monic polynomial and illustrate with numerical examples that the bounds are better than the existing ones.

We study the stability of certain spectra under some algebraic conditions weaker than the commutativity and we generalize many known commutative perturbation results. In particular, in a complex unital Banach algebra $$\mathcal {A},$$ we prove that if $$x \in \hbox {comm}_{w}(a)$$ and a is nilpotent, then $$\sigma (x)=\sigma (x+a).$$ Among other things, we prove that if $$x \in \hbox {comm}(xy)\cup \hbox {comm}(yx)$$ or $$y\in \hbox {comm}(yx)$$ for all $$x,y \in \mathcal {A},$$ then $$\mathcal {A}$$ is commutative. When $$\mathcal {A}=L(X)$$ is the unital complex Banach algebra of bounded linear operators acting on the complex Banach space X,  then $$\sigma _{p}(T){{\setminus }}\{0\}\subset \sigma _{p}(T+N){\setminus }\{0\},$$ where N is nilpotent and $$T\in \hbox {comm}(NT).$$

For $$x\in A\cup B,$$ define $$d_n=d\left( T^nx,T^{n-1}x\right) , n\ge 1$$ where A and B are subsets of a metric space and T is a cyclic map on $$A\cup B$$ . In this paper, we introduce a new class of mappings called cyclic uniform Lipschitzian mappings for which $$\{d_n\}$$ is not necessarily a non-increasing sequence and therein prove the existence of a best proximity pair. We also introduce a notion called proximal uniform normal structure and using the same we prove the existence of a best proximity pair for such mappings. Some open problems in this direction are also discussed.

In this paper, we show that every function in an anisotropic fractional Sobolev space defined on a regular domain extends to a function defined on the whole $${\mathbb {R}}^n$$ with the same Sobolev indices. As an application, we get that such functions are Hölder continuous and we give an explicit estimate for the continuity exponent.

In this paper, we improve the order of approximation of certain operators linking Bernstein and genuine Bernstein–Durrmeyer operators. Firstly, we obtain some direct results in terms of modulus of continuity and Voronovskaja type asymptotic formula for these operators. Finally, we give some numerical examples regarding the obtained theoretical results for new constructed operators.