Lectures on Hyponormal Operators

Abstract. Associated with T = U|T| (polar decomposition) in L(H) is a related operatorT = |T|Let H be a complex Hilbert space, and denote by L(H) the algebra of all bounded linear operators on H. If T ∈ L(H), we write σ (T), σ ap (T), and σ p (T) for the spectrum, the approximate point spectrum, and the point spectrum of T, respectively.An arbitrary operator T ∈ L(H) has a unique polar decomposition T = U|T|, where |T| = (T * T) 1 2 and U is the appropriate partial isometry satisfying kerU = ker|T| = kerT and kerU * = kerT * . Associated with T is a related operator |T| 1 2 U|T|

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“…IfT is decomposable, it has Dunford's property (C) from [8]. Then T has Dunford's property (C), by Theorem 1.12.…”

Section: Ift Is Decomposable Then T Is Quasidecomposablementioning

confidence: 91%

Some Connections Between an Operator and Its Aluthge Transform

Kim

Eungil

2005

Glasgow Math. J.

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“…In two dimensions, with R 2 identified with the complex plane C in the usual way, the exponential transform of a domain Ω has previously [19], [10], [17] been defined in polarized form as…”

Section: On the Two-dimensional Exponential Transform And Analytic Comentioning

confidence: 99%

“…The two-dimensional statements (8.4), (8.3) are in principle well-known and consequences of the fact that the polarized interior exponential transform H Ω (z, w) is a positive semidefinite kernel [5], [17]. Defining…”

Section: Remark 82mentioning

confidence: 99%

“…The exponential transform (in the terminology adopted throughout this paper) or the related integral transform of a domain in the complex plane has proved to be a remarkable object within branches of mathematics such as function theory [4], [13], operator theory [18], [5], see also [17], moment problems [19], [20], inverse recovery problems [12], analytic continuation [11], free boundary problems [10] and so forth. A corresponding exponential transform on the real axis was already known and used by A.A. Markov (in the 19th century) and later by N.I.…”

Section: Introductionmentioning

confidence: 99%

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The exponential transform: a renormalized Riesz potential at critical exponent

Gustafsson¹,

Putinar²

2003

Indiana Univ. Math. J.

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“…An operator T on a Hilbert space is called hyponormal if its self-commutator [T * , T ] = T * T − T T * is a positive operator [96]. Thirty years ago R. W. Carey and J. D. Pincus [18] found that if [T, T * ] moreover has rank one then T can be characterized up to unitary equivalence by a function 0 ≤ ρ ≤ 1, called the principal function and related to T by…”

Section: Operator Theory and The Exponential Transformmentioning

confidence: 99%