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On the extended spectrum of some quasinormal operators

In this paper we consider some extended eigenvalue problems for some quasinormal operators. The spectrum of an algebra homomorphism defined by a compact normal operator is also investigated. | Turk J Math (2017) 41: 1477 – 1481 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1610-99 Research Article On the extended spectrum of some quasinormal operators 1 Meltem SERTBAS ¸ 1,∗, Fatih YILMAZ2 Department of Mathematics, Faculty of Science, Karadeniz Technical University, Trabzon, Turkey 2 Institute of Natural Sciences, Karadeniz Technical University, Trabzon, Turkey Received: 26.10.2016 • Accepted/Published Online: 26.01.2017 • Final Version: 23.11.2017 Abstract: In this paper we consider some extended eigenvalue problems for some quasinormal operators. The spectrum of an algebra homomorphism defined by a compact normal operator is also investigated. Key words: Quasinormal operators, extended eigenvalue, extended spectrum 1. Introduction Let H be an infinite separable complex Hilbert space and denote by L(H) the set of bounded linear operators on H . A complex number λ is said to be an extended eigenvalue of a bounded operator A if there exists a nonzero operator T such that T A = λAT. T is called a λ eigenoperator for A and the set of extended eigenvalues is represented by σext (A). This condition takes place in quantum mechanics and analysis for their spectra [6]. Moreover, there is a nonzero operator Y such that XA = AY (1.1) and εA is the set of all X satisfying (1.1), and then it is easily seen that εA is an algebra. When A has dense range, one can define the map ΦA : εA → L (H) by ΦA (X) = Y and verify that ΦA is an algebra homomorphism. This homomorphism is a closed (generally unbounded) linear transformation. Biswas et al. defined an eigenvalue of ΦA as an extended eigenvalue of A and proved that the set of extended eigenvalues of the Volterra operator V is equal to the interval (0, +∞) in [2]. Karaev gave the set of extended eigenvectors of the Volterra operator V on L2 [0, 1] in [11]. However, the problem is open as to the other spectrum parts of ΦV . Furthermore, Biswas and .