Contractions without non-trivial invariant subspaces satisfying a positivity condition

Abstract

Abstract An operator A ∈ B ( H ) $A\in B(\mathcal{H})$ , the algebra of bounded linear transformations on a complex infinite dimensional Hilbert space H $\mathcal{H}$ , belongs to class A ( n ) $\mathcal{A}(n)$ (resp., A ( ∗ − n ) $\mathcal{A}(*-n)$ ) if | A | 2 ≤ | A n + 1 | 2 n + 1 $\vert A\vert^{2}\leq\vert A^{n+1}\vert^{\frac{2}{n+1}}$ (resp., | A ∗ | 2 ≤ | A n + 1 | 2 n + 1 $\vert A^{*}\vert^{2}\leq \vert A^{n+1}\vert^{\frac{2}{n+1}}$ ) for some integer n ≥ 1 $n\geq1$ , and an operator A ∈ B ( H ) $A\in B(\mathcal{H})$ is called n-paranormal, denoted A ∈ P ( n ) $A\in \mathcal{P}(n)$ (resp., ∗ − n $*-n$ -paranormal, denoted A ∈ P ( ∗ − n ) $A\in \mathcal{P}(*-n)$ ) if ∥ A x ∥ n + 1 ≤ ∥ A n + 1 x ∥ ∥ x ∥ n $\Vert Ax\Vert ^{n+1}\leq \Vert A^{n+1}x\Vert \Vert x\Vert ^{n}$ (resp., ∥ A ∗ x ∥ n + 1 ≤ ∥ A n + 1 x ∥ ∥ x ∥ n $\Vert A^{*}x\Vert ^{n+1}\leq \Vert A^{n+1}x\Vert \Vert x\Vert ^{n}$ ) for some integer n ≥ 1 $n\geq 1$ and all x ∈ H $x \in\mathcal{H}$ . In this paper, we prove that if A ∈ { A ( n ) ∪ P ( n ) } $A\in\{\mathcal{A}(n)\cup \mathcal{P}(n)\}$ (resp., A ∈ { A ( ∗ − n ) ∪ P ( ∗ − n ) } $A\in\{\mathcal{A}(*-n)\cup \mathcal{P}(*-n)\}$ ) is a contraction without a non-trivial invariant subspace, then A, | A n + 1 | 2 n + 1 − | A | 2 $\vert A^{n+1}\vert^{\frac{2}{n+1}}-\vert A\vert^{2}$ and | A n + 1 | 2 − n + 1 n | A | 2 + 1 $\vert A^{n+1}\vert^{2}- {\frac{n+1}{n}}\vert A\vert^{2}+ 1$ (resp., A, | A n + 1 | 2 n + 1 − | A ∗ | 2 $\vert A^{n+1}\vert^{\frac{2}{n+1}}-\vert A^{*}\vert^{2}$ and | A n + 2 | 2 − n + 1 n | A | 2 + 1 ≥ 0 $\vert A^{n+2}\vert^{2}- {\frac{n+1}{n}}\vert A\vert^{2}+ 1\geq0$ ) are proper contractions.

Keywords

• class A ( n ) $\mathcal{A}(n)$ operator
• class A ( ∗ − n ) $\mathcal{A}(*-n)$ operator
• P ( n ) $\mathcal{P}(n)$ operator
• class P ( ∗ − n ) $\mathcal{P}(*-n)$ operator
• contraction
• proper contraction