Let Ψn(x) be the monic polynomial having precisely all non-primitive nth roots of unity as its simple zeros. One has Ψn(x) = (x n − 1)/Φn(x), with Φn(x) the nth cyclotomic polynomial. The coefficients of Ψn(x) are integers that like the coefficients of Φn(x) tend to be surprisingly small in absolute value, e.g. for n < 561 all coefficients of Ψn(x) are ≤ 1 in absolute value. We establish variou...

This paper discusses several structural and fundamental properties of the $k^{th}$-order slant Toeplitz operators on Lebesgue space $n$- torus $\mathbb{T}^n$, for integers $k\geq 2$ $n\geq 1$. We obtain certain equivalent conditions commutativity essential these operators. In last section, we deal with spectrum a operator $L^2(\mathbb{T}^n)$ investigate such an to be isometry, hyponormal or nor...

this paper presents characterizations of m-fuzzifying matroids bymeans of two kinds of fuzzy operators, called m-fuzzifying derived operatorsand m-fuzzifying difference derived operators.

In this note we prove that if A and B∗ are subnormal operators and X is a bounded linear operator such that AX − XB is a Hilbert-Schmidt operator, then f(A)X −Xf(B) is also a Hilbert-Schmidt operator and ||f(A)X −Xf(B)||2 ≤ L ||AX −XB||2, for f belonging to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and X ∈ L(H) i...

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 ...

We prove that if A and B∗ are subnormal operators and X is a bounded linear operator such that AX −XB is a Hilbert-Schmidt operator, then f A X −Xf B is also a Hilbert-Schmidt operator and ‖f A X −Xf B ‖2 ≤ L‖AX −XB‖2 for f belongs to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and X ∈ L H is such that SX − XT belo...

In this paper we deal with the subnormality and the quasinormality of Toeplitz operators with matrix-valued rational symbols. In particular, in view of Halmos’s Problem 5, we focus on the question: Which subnormal Toeplitz operators are normal or analytic ? We first prove: Let Φ ∈ L∞Mn be a matrix-valued rational function having a “matrix pole,” i.e., there exists α ∈ D for which kerHΦ ⊆ (z−α)H...

A set of real nth roots that is pairwise linearly independent over the rationals must also be linearly independent. We show how this result may be extended to more general fields.

Abstract. For 2-variable weighted shifts W(α,β) ≡ (T1, T2) we study the invariance of (joint) khyponormality under the action (h, `) 7→ W (h,`) (α,β) := (T h 1 , T ` 2 ) (h, ` ≥ 1). We show that for every k ≥ 1 there exists W(α,β) such that W (h,`) (α,β) is k-hyponormal (all h ≥ 2, ` ≥ 1) but W(α,β) is not k-hyponormal. On the positive side, for a class of 2-variable weighted shifts with tensor...