A note on quasisimilarity. II

Fialkow, Lawrence A.

doi:10.2140/pjm.1977.70.151

Cited by 14 publications

(11 citation statements)

References 5 publications

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“…Evidently, Theorem 2.4 implies, in particular, that quasi-similarity preserves the spectrum for quotients and restrictions of decomposable operators. This improves a number of previous results in this direction and covers, for instance, the case of hyponormal and cohyponormal operators, see [9], [13], [16] for further information. Moreover, it follows that the same kind of spectral invariance holds for the weaker Another natural example arises in harmonic analysis, where the Fourier transform acts as an injective intertwiner for a given convolution operator and the corresponding multiplication operator.…”

Section: -(T C (T*j*)(a*) £ O-c (St)(a)supporting

confidence: 77%

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Local spectral theory and spectral inclusions

Laursen

Neumann

1994

Glasgow Math. J.

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Introduction. Suppose that T and 5 are continuous linear operators on complex Banach spaces X and Y, respectively, and that A is a non-zero continuous linear mapping from X to Y. If A intertwines T and 5 in the sense that SA = A T, then a classical result due to Rosenblum implies that the spectra cr(T) and cr(S) must overlap, see [12]. Actually, Davis and Rosenthal [5] have shown that the surjectivity spectrum cr su (r) will meet the approximate point spectrum o" ap (S) in this case (terms to be denned below). Further information about the relations between the two spectra and their finer structure becomes available when the intertwiner A is injective or has dense range, see

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Section: -(T C (T*j*)(a*) £ O-c (St)(a)supporting

confidence: 77%

“…S T on a Hilbert space such that A and A* are injective, SA = AT, S is quasi-nilpotent, and the spectrum of T is the unit disc, see for instance [9] or [13]. Obviously, in this case, none of the sets 0-^(7), a su (T), (r ap (T*), a su (T*) is contained in cr(S) = cr(S*).…”

Section: ({0})<=$ S (T C (A))mentioning

confidence: 99%

Local spectral theory and spectral inclusions

Laursen

Neumann

1994

Glasgow Math. J.

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“…Let Wa and WB be the bilateral weighted shifts with weight sequences {an}™=_a0 and {/5B}"__00, respectively. L. A. Fialkow shows in [6] that Wa and Wß are quasisimilar but not similar. Note that Wa is cohyponormal.…”

Section: Proof L a Fialkow Showed In [6] That If Either A Or B Is mentioning

confidence: 99%

“…We shall need the following lemma. [6] that any two quasisimilar injective bilateral weighted shifts have equal spectra.) Since hyponormal operators are spectraloid, we have \\A\\ = ||Ä||.…”

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confidence: 97%

Equality of essential spectra of certain quasisimilar seminormal operators

Williams¹

1980

Proc. Amer. Math. Soc.

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Abstract. Let A and B be quasisimilar seminormal operators on a separable, infinite dimensional complex Hilbert space. Several conditions which imply that A and B have equal essential spectra are presented. For example, if A and B are both biquasitriangular then A and B have equal essential spectra.Let DC denote a separable, infinite dimensional complex Hilbert space, and let £(DC) denote the algebra of all bounded linear operators on DC. (We shall use the term operator to mean an element of £(DC). Unless specified otherwise, all operators are assumed to be acting on the Hilbert space DC.) An operator T is said to be hyponormal if TT* < T* T. If T is an operator and T* is hyponormal, then T is said to be cohyponormal. If T is either hyponormal or cohyponormal, then T is called a seminormal operator. An operator X having trivial kernel and dense range is called a quasiaffinity. Operators A and B are said to be quasisimilar if there exist quasiaffinities X and Y such that XA = BX and AY = YB. If T belongs to £(DC), we shall let %(T) denote the kernel of T, a(T) the spectrum of T, and ae(T) the essential spectrum of T, i.e., the spectrum of w( T), where it is the natural quotient map of £(DC) onto the Calkin algebra £(DC)/G (6 denotes the norm-closed ideal of all compact operators in £(DC)).It is known that quasisimilar normal operators are unitarily equivalent (cf.[3]). Thus quasisimilar normal operators have equal spectra and essential spectra. S. Clary constructed an example in [2] which shows that quasisimilar hyponormal operators need not be similar. Nevertheless, he proved that quasisimilar hyponormal operators do have equal spectra. Clary's proof, with only modest modifications, shows that quasisimilar seminormal operators have equal spectra. In view of the above results, it is natural to pose the following question: Do quasisimilar seminormal operators have equal essential spectra? At the present we are unable to answer this question. However, in this note, we do show that the answer is "yes" in several interesting cases.Let A and B be operators and X a quasiaffinity satisfying XA = BX. M. Radjabalipour proved recently in [10] that if A * and B are hyponormal, then A and B are unitarily equivalent normal operators. (H. Radjavi and P. Rosenthal proved earlier in [11] that the above result is valid in the subnormal case.) Let A and B be quasisimilar seminormal operators. In view of Radjabalipour's result, in order to

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“…Recall ( [4], [14]) that an operator T 2 BðHÞ is said to be a quasi-affine transform of S 2 BðHÞ if there exists an injection X 2 BðHÞ with dense range such that SX ¼ XT, and this relation of T and S is denoted by T 0 S. If both T 0 S and S 0 T, then we say that T and S are quasi-similar. In general, quasi-similarity preserves the point spectrum but not the spectrum, the set of all eigenvalues of finite multiplicity, the set of all isolated eigenvalues, nor the set of all isolated points of the spectrum.…”

Section: Introduction Let H Be An Infinite Dimensional Complex Hilbementioning

confidence: 99%