ON QUADRATICALLY HYPONORMAL WEIGHTED SHIFTS (Structural study of operators via spectra or numerical ranges)
Mi Ryeong Lee · Institutional Repositories DataBase (IRDB) · 2012
The positive quadratic hyponormality are characterized and quadratic hyponormality of certain such backstep extensions of arbitrary length are generalized from some earlier results.These results can be applied to the positive quadratic hyponormality of the weighted shift $W_{\alpha}$ associated to the weight sequence $\alpha$ : 1, 1, $\sqrt{x},$ $(\sqrt{u}, \sqrt{v}, \sqrt{w})^{\wedge}$ .This yields the set of positive real numbers $x$ such that $W_{\alpha}$ is quadratically hyponormal for some $u,$ $v$ and $w$ , an open problem in [9], and one produces an interval in $x$ with nonempty interior in the positive real line for quadratic hyponormality but not positive quadratic hyponormality for such a weighted shift $W_{\alpha}$ .1. Introduction.This is based on the joint work with George R. Exner, Il Bong Jung, and Sun Hyun Park and was talked at the 2011 RIMS symposium: Structural study of operators via spectra or numerical ranges, which was held at Kyoto University on November 14-16 in 2011.And also this will be appeared in some other journal as a version with detail proofs and additional results (cf.[20]).Let $H$ be a separable complex Hilbert space and $L('H)$ be the algebra of all bounded linear operators on $H$ .An operator $T$ in $L('H)$ is normal if it commutes with its adjoint, subnormal if it is the restriction of a normal operator to an invariant subspace, and hyponormal if $T^{*}T\geq TT^{*}$ .For $A,$ $B\in L(\mathcal{H})$ , we set $[A, B]$ $:=$ AB-BA.A k-tuple $T=(T_{1}, \cdots, T_{k})$ of operators in $L(\mathcal{H})$ is called hyponormal if the operator matrix $([T_{j}^{*}, T_{i}])_{i,j=1}^{k}$ is positive on the direct sum of $k$ copies of $\mathcal{H}$ .For $k\in \mathbb{N}$ and $T\in L('H),$ $T$ is said to be k-hyponormal if $(I, T, \cdots, T^{k})$ is hyponormal.It is well-known that $T\in L(H)$ is subnormal if and only if $T$ is k-hyponormal for all $k\in \mathbb{N}$ , where $\mathbb{N}$ is the set of natural numbers ([14], [2]).A k-tuple $T=(T_{1}, \cdots, T_{k})$ is weakly hyponormal if $\lambda_{1}T_{1}+\cdots+\lambda_{k}T_{k}$ is hyponormal for every $\lambda_{i}\in \mathbb{C},$ $i=1,$ $\cdots,$ $k$ , where $\mathbb{C}$is the set of complex numbers.An operator $T$ is weakly k-hyponormal if $(T, T^{2}, \cdots, T^{k})$ is weakly hyponormal; equivalently, for every complex polynomial $p$ of degree $k$ or less, $p(T)$ is hyponormal ([4]).An operator $T$ is polynomially hyponormal if, for every polynomial $p$ with complex coefficients, $p(T)$ is hyponormal.In [4] Curto initiated the study of classes (actually or potentially) between hyponormal and subnormal with the implications: $subnormal\Rightarrow\cdots\Rightarrow 2-$ hyponormal $\Rightarrow$ hyponormal"; the converse is not always true ([5]).Also it holds obviously that $subnormal\Rightarrow$ polynomially hyponormal $\Rightarrow\cdots\Rightarrow$ weakly $2-hyponormal\Rightarrow$ hyponormal"; but the converse implications are not developed completely yet except for weak 2-hyponormalities ([11]).In particular, the case in which $k=2$ has received considerable attention (see, for '2000 Mathematics Subject Classification.Primary $47B37$ ; Secondary $47B20$ .