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On $n$ $n$ -quasi- $m$ $m$ -isometric operators

College of Science, Department of Mathematics, Taibah University, P. O. Box 20003, Al Madinah Al Munawarah, Saudi Arabia

E-mail Address: mecherisalah@hotmail.com

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and

T. Prasad

The Institute of Mathematical Sciences, CIT Campus, Chennai-600 113, India

E-mail Address: prasadvalapil@gmail.com

Corresponding author.

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https://doi.org/10.1142/S179355711650073XCited by:23 (Source: Crossref)

Abstract

We introduce the class of $n$ $n$ -quasi- $m$ $m$ -isometric operators on Hilbert space. This generalizes the class of $m$ $m$ -isometric operators on Hilbert space introduced by Agler and Stankus. An operator $T \in B (H)$ $T \in B (H)$ is said to be $n$ $n$ -quasi- $m$ $m$ -isometric if

T * n (m \sum k = 0 (- 1) k (m k) T * m - k T m - k) T n = 0 . <math display="block" altimg="eq-00011.gif"><mrow><msup><mrow><mi>T</mi></mrow><mrow><mo stretchy="false" class="MathClass-bin">*</mo><mi>n</mi></mrow></msup><mfenced separators="" open="(" close=")"><mrow><munderover accentunder="true" accent="true"><mrow><mo class="MathClass-op">\sum</mo></mrow><mrow><mi>k</mi><mo class="MathClass-rel">=</mo><mn>0</mn></mrow><mrow><mi>m</mi></mrow></munderover><msup><mrow><mo class="MathClass-open" stretchy="false">(</mo><mo stretchy="false" class="MathClass-bin">-</mo><mn>1</mn><mo class="MathClass-close" stretchy="false">)</mo></mrow><mrow><mi>k</mi></mrow></msup><mfenced separators="" open="(" close=")"><mrow><mtable style="text-align:axis" equalrows="false" equalcolumns="false" class="array"><mtr><mtd class="array" columnalign="center"><mi>m</mi></mtd></mtr><mtr><mtd class="array" columnalign="center"><mi>k</mi></mtd></mtr></mtable></mrow></mfenced><msup><mrow><mi>T</mi></mrow><mrow><mo stretchy="false" class="MathClass-bin">*</mo><mi>m</mi><mo stretchy="false" class="MathClass-bin">-</mo><mi>k</mi></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mi>m</mi><mo stretchy="false" class="MathClass-bin">-</mo><mi>k</mi></mrow></msup></mrow></mfenced><msup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msup><mo class="MathClass-rel">=</mo><mn>0</mn><mo class="MathClass-punc">.</mo></mrow></math>

In this paper

$2 \times 2$ matrix representation of a

$n$ -quasi-

$m$ -isometric operator is given. Using this representation we establish some basic properties of this class of operators.

Communicated by W. C. Wong

Keywords:

AMSC: 47A62, 47A63, 47A10

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