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Published in:
2009 | OriginalPaper | Chapter
Trace Ideals
Author : Rajendra Bhatia
Published in: Notes on Functional Analysis
Publisher: Hindustan Book Agency
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Let A be a compact operator on (an infinite-dimensional) Hilbert space ℋ and let (23.1)<math display='block'> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> <mrow><mo>(</mo> <mi>A</mi> <mo>)</mo></mrow><mo>≥</mo><msub> <mi>s</mi> <mn>2</mn> </msub> <mrow><mo>(</mo> <mi>A</mi> <mo>)</mo></mrow><mo>≥</mo><mo>⋯</mo><mo>≥</mo><mn>0</mn> </mrow> </math> $${s_1}\left( A \right) \geqslant {s_2}\left( A \right) \geqslant \cdots \geqslant 0$$ be the singular values of A. The sequence s n (A) converges to 0. In this lecture we study special compact operators for which this sequence belongs to the space ℓ1 or the space ℓ2.