On Quasisimilarity for Log-Hyponormal Operators

Jeon, In-Ho; Tanahashi, Kôtarô; Uchiyama, Atsushi

doi:10.1017/s0017089503001642

Cited by 14 publications

(5 citation statements)

References 17 publications

(19 reference statements)

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“…There are some significant evidences for this assertion, for instance, it is proved in [14] that any operator A has a nontrivial invariant subspace if and only if so does Ã. Another interesting application deals with an application of the Aluthge transform for generalizing the Fuglede-Putnam theorem [12]. It indeed is a motivation for our work in this paper.…”

Section: Introductionmentioning

confidence: 84%

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Fuglede–Putnam type theorems via the Aluthge transform

Moslehian

Sales

2012

Positivity

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Let A = U |A| and B = V |B| be the polar decompositions of A ∈ B(H 1 ) and B ∈ B(H 2 ) and let Com(A, B) stand for the set of operators X ∈ B(H 2 , H 1 ) such that AX = XB. A pair (A, B) is said to have the FP-property if Com(A, B) ⊆ Com(A * , B * ). LetC denote the Aluthge transform of a bounded linear operator C. We show that (i) if A and B are invertible and (A, B) has the FP-property, then so is (Ã,B); (ii) if A and B are invertible, the spectrums of both U and V are contained in some open semicircle and (Ã,B) has the FP-property, then so is (A, B); (iii) if (A, B) has the FP-property, then Com(A, B) ⊆ Com(Ã,B), moreover, if A is invertible, then Com(A, B) = Com(Ã,B). Finally, if Re(U |A| 1 2 ) ≥ a > 0 and Re(V |B| 1 2 ) ≥ a > 0 and X is an operator such that U * X = XV , then we prove that Ã * X − XB p ≥ 2a |B| 1 2 X − X|B| 1 2 p for any 1 ≤ p ≤ ∞.2010 Mathematics Subject Classification. Primary 47B20; Secondary 47B15, 47A30.

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Section: Introductionmentioning

confidence: 84%

“…[12] Let A ∈ B(H 1 ) and B * ∈ B(H 2 ) be either log-hyponormal or phyponormal operators. Then the pair(A, B)has the FP-property.…”

mentioning

confidence: 99%

Fuglede–Putnam type theorems via the Aluthge transform

Moslehian

Sales

2012

Positivity

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“…be the polar decompositions of T 2 and S Ã 2 , respectively and Conway [10] proved that the normal parts of quasisimilar subnormal operators are unitarily equivalent and gave an example showing that the pure parts of quasisimilar subnormal operators need not be quasisimilar. This result was gener-alized to classes of p-hyponormal operators in [21] and log-hyponormal operators in [23], respectively. We prove that these results hold for class wAðs; tÞ operators with s þ t ¼ 1.…”

Section: Class Wa(s T) Operators and Quasi-similaritymentioning

confidence: 99%

Class $wA(s,t)$ operators and quasisimilarity

Rashid¹

2013

Port. Math.

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In this paper it is shown that the normal parts of quasisimilar wAðs; tÞ operators with s þ t ¼ 1 are unitarily equivalent. Also, we establish the orthogonality of the range and the kernel of a nonnormal derivation with respect to the unitarily invariant norms associated with norm ideals of operators. Moreover, we obtain that the range of the generalized derivation induced by an pair satisfies Fuglede-Putnam property is orthogonal to its kernel.

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“…An invertible p-hyponormal operator is log-hyponormal, but the converse is false; see [17, p. 169] for a reference. Log-hyponormal and p-hyponormal operators, which share a number of properties with hyponormal operators, have been considered by a number of authors in the recent past; see [11,17,24] for further references.…”

Section: −Hyponormal Operator Is Semihyponormal) It Is An Immediatmentioning

confidence: 99%

Ranges and kernels of derivations

Ech-chad¹

2017

Turk J Math

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In this paper we establish some properties concerning the class of operators A ∈ L(H) that satisfy R(δA) ∩ {A} ′ = {0} , where R(δA) is the norm closure of the range of the inner derivation δA, defined on L(H) by δA(X) = AX −XA . Here H stands for a Hilbert space; as a consequence, we show that the set {A ∈ L(H) / R(δA)∩{A} ′ = {0}} is norm-dense. We also describe some classes of operators A, B for which we have R(δA,B) ∩ ker(δA * ,B * ) = {0} ( ker(δA * ,B * ) is the kernel of the generalized derivation δA * ,B * defined on L(H) by δA * ,B * (X) = A * X − XB * ).

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On Quasisimilarity for Log-Hyponormal Operators

Cited by 14 publications

References 17 publications

Fuglede–Putnam type theorems via the Aluthge transform

Fuglede–Putnam type theorems via the Aluthge transform

Class $wA(s,t)$ operators and quasisimilarity

Ranges and kernels of derivations

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