Journal of Mathematical Analysis ISSN: 2217-3412, URL: http://www.ilirias.com Volume 6 Issue 2(2015), Pages 13-21. SOME PROPERTIES OF (p, k)−QUASIPOSINORMAL OPERATORS N.L. BRAHA, ILMI HOXHA, KOTARO TANAHASHI Abstract. An operator T ∈L(H) is said to be a (p, k)−quasiposinormal operator if T *k ( c 2 (T * T ) p − (TT * ) p ) T k ≥ O for some c> 0, for a positive integer k and a positive number 0 <p ≤ 1. In this paper we show some properties of (p, k)−quasiposinormal operators, Fuglede−Putnam Theorem in the case where X is a Hilbert−Schmidt op- erator, A is a (p, k)−quasiposinormal operator with c> 0, and (B -1 ) * is a(p, k)−quasiposinormal operator with d> 0 such that AX = XB and ‖|A * | 1-p ‖‖|B -1 | 1-p ‖≤ 1 cd , then A * X = XB * . Also, we consider tensor products for (p, k)−quasiposinormal operators. 1. Introduction Throughout this paper, let H and K be infinite dimensional separable complex Hilbert spaces with inner product 〈·, ·〉. We denote by L(H, K) the set of all bounded operators from H into K. To simplify, we put L(H) := L(H, H). For T ∈L(H), we denote by kerT the null space and by T (H) the range of T . The null operator and the identity on H will be denoted by O and I , respectively. If T is an operator, then T * is its adjoint, and ‖T ‖ = ‖T * ‖. We shall denote the set of all complex numbers by C and the complex conjugate of a complex number λ by λ. The closure of a set M will be denoted by M and we shall henceforth shorten T − λI to T − λ. For an operator T ∈L(H), as usual, |T | =(T * T ) 1 2 . An operator T ∈L(H) is said to be hyponormal if |T | 2 ≥|T * | 2 . An operator T ∈L(H) is said to be paranormal if ‖T 2 x‖≥‖Tx‖ 2 for any unit vector x in H.T is said to be log−hyponormal if T is invertible and satisfies the following inequality log(T * T ) ≥ log(TT * ) [22]. An operator T is said to be dominant in the sense of J. G. Stampfli and B. L. Wadhwa [21] if ran(T − λ) ⊆ ran(T − λ) * for all complex number λ or, equivalently, if there is a real number M λ ≥ 1 such that ‖(T − λ) * x‖≤ M λ ‖(T − λ)x‖ for all x ∈H. An operator T belongs to class Y if there exist α ≥ 1 and k α > 0 such that |TT * − T * T | α ≤ k 2 α (T − λ) * (T − λ) for all λ ∈ C. 2010 Mathematics Subject Classification. 47A10, 47B37, 15A18. Key words and phrases. (p, k)−quasiposinormal operator, Fuglede−Putnam theorem, tensor product. c 2015 Universiteti i Prishtin¨ es, Prishtin¨ e, Kosov¨ e. Submitted April 11, 2015. Published May 20, 2015. 13