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In this paper, we introduce some explicit and recursive formulas for the n$$ n $$th q$$ q $$‐derivative of a quotient of two functions. We establish the connections between linear algebra, the homomorphisms between commutative algebras and the properties of q$$ q $$‐differential operator. One of the formulas involves the determinant of the functions and their q$$ q $$‐derivatives. Then, we show that the formulas in the classical mathematical analysis are their special cases. Finally, we used formulas to obtain some interesting q$$ q $$‐identities. [ABSTRACT FROM AUTHOR]

We study the star and minus partial orders on the set $ {\mathcal {B}}(H,K) $ B (H , K) of all bounded operators acting from a Hilbert space H to a Hilbert space K. We extend and strengthen some results from matrix case to the case of general operators, which may not possess generalized inverses. By means of some norm inequalities, we give necessary and sufficient condition under which two operators have orthogonal ranges, and thus, we give a characterization of the star partial order. When A = PB for some projection P, we prove the equivalence of $ A \lt ^*B $ A < ∗ B with $ f(AA^*)A \lt ^*f(BB^*)B $ f (A A ∗) A < ∗ f (B B ∗) B for a wide class of continuous functions f. Also, we prove that $ A \lt ^*B $ A < ∗ B if and only if $ A \lt ^-B $ A < − B and $ A^2 \lt ^{-}B^{2} $ A 2 < − B 2 when A is a weak EP operator and B is a self-adjoint operator. Finally, we consider the Moore–Penrose invertibility and the ordinary invertibility of a linear combination of operators when they are related with one of these two orders. [ABSTRACT FROM AUTHOR]

A positive answer on a question posed in Eshkaftaki et al. [DSS-weak majorization and its linear preservers on ℓ p spaces. Linear Multilinear Algebra. 2018;66(10):2076–2088] is provided. Precisely, the concrete form of bounded linear operators which preserve DSS-weak majorization on discrete Lebesgue space ℓ 1 (I) , when I is an infinite set, is presented. Interestingly, it is shown that the class of DSS-weak majorization preservers is a proper subset of the class of majorization preservers on ℓ 1 (I) , despite the fact that these two classes of preservers coincide [Eshkaftaki et al. DSS-weak majorization and its linear preservers on ℓ p spaces. Linear Multilinear Algebra. 2018;66(10):2076–2088] when above relations are defined on ℓ p (I) when p ∈ (1 , ∞). In this way, the conjecture given at the end of the paper [Eshkaftaki et al. DSS-weak majorization and its linear preservers on ℓ p spaces. Linear Multilinear Algebra. 2018;66(10):2076–2088] is confirmed. [ABSTRACT FROM AUTHOR]

Letbe a unital ring admitting involution. We introduce an order onand show that in the case whenis a Rickart-ring, this order is equivalent to the well-known star partial order. The notion of the left-star and the right-star partial orders is extended to Rickart-rings. Properties of the star, the left-star and the right-star partial orders are studied in Rickart-rings and some known results are generalized. We found matrix forms of elementsandwhen, whereis one of these orders. Conditions under which these orders are equivalent to the minus partial order are obtained. The diamond partial order is also investigated. [ABSTRACT FROM AUTHOR]