A note on $”nth$ roots of operators”

A note on $lambda$-Aluthge transforms of operators

Let $A=U|A|$ be the polar decomposition of an operator $A$ on a Hilbert space $mathscr{H}$ and $lambdain(0,1)$. The $lambda$-Aluthge transform of $A$ is defined by $tilde{A}_lambda:=|A|^lambda U|A|^{1-lambda}$. In this paper we show that emph{i}) when $mathscr{N}(|A|)=0$, $A$ is self-adjoint if and only if so is $tilde{A}_lambda$ for some $lambdaneq{1over2}$. Also $A$ is self adjoint if and onl...

A Note on Weighted Composition Operators on L^p-Spaces

On the integrality of nth roots of generating functions

Motivated by the discovery that the eighth root of the theta series of the E8 lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f ∈ R (where R = 1 + xZ x ) can be written as f = gn for g ∈ R, n 2. Let Pn := {gn | g ∈R} and let μn := n∏p |n p. We show among other things that (i) for f ∈R, f ...

Polynomials that Vanish on Distinct nth Roots of Unity

Let C denote the ̄eld of complex numbers and n the set of nth roots of unity. For t = 0; : : : ; n ¡ 1, de ̄ne the ideal =(n; t +1) 1⁄2 C [x0; : : : ; xt] consisting of those polynomials in t + 1 variables that vanish on distinct nth roots of unity; that is, f 2=(n; t+ 1) if and only if f (!0 ; : : : ; !t) = 0 for all (!0 ; : : : ; !t) 2 t+1 n satisfying !i 6= !j, for 0 · i < j · t. In this paper...

A Note on Quadratic Maps for Hilbert Space Operators

In this paper, we introduce the notion of sesquilinear map on Β(H) . Based on this notion, we define the quadratic map, which is the generalization of positive linear map. With the help of this concept, we prove several well-known equality and inequality...