Inverse functions of polynomials and orthogonal polynomials as operator monotone functions

Abstract

Abstract. We study the operator monotonicity of the inverse of every polynomial with a positive leading coefficient. Let {pn} ∞ n=0 be a sequence of orthonormal polynomials and p n+ the restriction of pn to [an, ∞), where an is the maximum zero of pn. Then p

Positive coincidence points for a class of nonlinear operators and their applications to matrix equations

“…where Q is a positive semi definite matrix, A i are arbitrary square matrices, f i are Löwner-Heinz monotone operators and g is an Uchiyama operator [16][17][18]. Our results improve, extend and generalize some existing ones in the literature [11,15,19,20].…”

“…In this section, we show at first that some existing monotone operators in the literature are already strictly super-homogeneous and surjective. Next, we derive a corollary from a previous result and solve nonlinear equations involving Uchiyama's operator functions [16,17]. We construct two algorithms for monotone and antitone operators, and we apply them to solve several nonlinear matrix equations.…”

“…(Löwner-Heinz inequality). The operator function t α is monotone for all α (Uchiyama[16,17]) Let g x 1 ( ) and g x 2 ( ) be functions defined on 0, . Then, the inverse of the operator functions g 1 and g 2 are operator monotone functions on 0, [ ∞).Lemma 5.3.…”

A new majorization between functions, polynomials, and operator inequalities II

“…That p À1 is semi-operator monotone on ½0; 1Þ, i.e. pð ffiffi t p Þ 2 2 P À1 þ ½0; 1Þ, has been shown in Theorem 4.1 of [10]. However we now give a simple and direct proof of it by making use of Product Theorem.…”

“…In Theorem 3.5 we will generalize it. To do so, we need the following lemma, whose first assertion was stated in Remark 1 of [9] and the second one was essentially proved in the proof of Theorem 3.2 of [10]. But for the sake of completeness we give a proof.…”