On Positive Quadratic Hyponormality of a Unilateral Weighted Shift with Recursively Generated by Five Weights

• Main Authors: Chunji Li, Cheon Seoung Ryoo
• Format: Article
• Language: English
• Published: MDPI AG 2019-02-01
• Series: Mathematics
• Subjects: positively quadratically hyponormal; quadratically hyponormal; unilateral weighted shift; recursively generated
• Online Access: https://www.mdpi.com/2227-7390/7/2/212

Let <inline-formula> <math display="inline"> <semantics> <mrow> <mn>1</mn> <mo>&lt;</mo> <mi>a</mi> <mo>&lt;</mo> <mi>b</mi> <mo>&lt;</mo> <mi>c</mi> <mo>&lt;</mo> <mi>d</mi> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mover accent="true"> <mi>&#945;</mi> <mo>^</mo> </mover> <mfenced open="[" close="]"> <mn>5</mn> </mfenced> </msub> <mo>:</mo> <mo>=</mo> <msup> <mfenced separators="" open="(" close=")"> <mn>1</mn> <mo>,</mo> <msqrt> <mi>a</mi> </msqrt> <mo>,</mo> <msqrt> <mi>b</mi> </msqrt> <mo>,</mo> <msqrt> <mi>c</mi> </msqrt> <mo>,</mo> <msqrt> <mi>d</mi> </msqrt> </mfenced> <mo>&and;</mo> </msup> </mrow> </semantics> </math> </inline-formula> be a weighted sequence that is recursively generated by five weights <inline-formula> <math display="inline"> <semantics> <mrow> <mn>1</mn> <mo>,</mo> <msqrt> <mi>a</mi> </msqrt> <mo>,</mo> <msqrt> <mi>b</mi> </msqrt> <mo>,</mo> </mrow> </semantics> </math> </inline-formula><inline-formula> <math display="inline"> <semantics> <mrow> <msqrt> <mi>c</mi> </msqrt> <mo>,</mo> <msqrt> <mi>d</mi> </msqrt> </mrow> </semantics> </math> </inline-formula>. In this paper, we give sufficient conditions for the positive quadratic hyponormalities of <inline-formula> <math display="inline"> <semantics> <msub> <mi>W</mi> <mrow> <mi>&#945;</mi> <mfenced open="(" close=")"> <mi>x</mi> </mfenced> </mrow> </msub> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msub> <mi>W</mi> <mrow> <mi>&#945;</mi> <mfenced separators="" open="(" close=")"> <mi>y</mi> <mo>,</mo> <mi>x</mi> </mfenced> </mrow> </msub> </semantics> </math> </inline-formula>, with <inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#945;</mi> <mfenced open="(" close=")"> <mi>x</mi> </mfenced> <mo>:</mo> <msqrt> <mi>x</mi> </msqrt> <mo>,</mo> <msub> <mover accent="true"> <mi>&#945;</mi> <mo>^</mo> </mover> <mfenced open="[" close="]"> <mn>5</mn> </mfenced> </msub> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#945;</mi> <mfenced separators="" open="(" close=")"> <mi>y</mi> <mo>,</mo> <mi>x</mi> </mfenced> <mo>:</mo> <msqrt> <mi>y</mi> </msqrt> <mo>,</mo> <msqrt> <mi>x</mi> </msqrt> <mo>,</mo> <msub> <mover accent="true"> <mi>&#945;</mi> <mo>^</mo> </mover> <mfenced open="[" close="]"> <mn>5</mn> </mfenced> </msub> <mo>.</mo> </mrow> </semantics> </math> </inline-formula>