On Subscalarity of Some 2 × 2 M-Hyponormal Operator Matrices

and Applied Analysis 3 Proof. Since T is anM-hyponormal operator, it follows from Corollary 2 that there exists a constant C D such that 󵄩󵄩󵄩󵄩󵄩 (I − P) ∂ i f n 󵄩󵄩󵄩󵄩󵄩2,D ≤ MC D ( 󵄩󵄩󵄩󵄩󵄩 (T − z) ∂ i+1 f n 󵄩󵄩󵄩󵄩󵄩2,D + 󵄩󵄩󵄩󵄩󵄩 (T − z) ∂ i+2 f n 󵄩󵄩󵄩󵄩󵄩2,D ) (17) for i = 0, 1, 2, . . . , m − 2. From (17), we have lim n→∞ 󵄩󵄩󵄩󵄩󵄩 (I − P) ∂ i f n 󵄩󵄩󵄩󵄩󵄩2,D = 0 (18) for i = 0, 1, 2, . . . , m − 2. Hence, lim n→∞ 󵄩󵄩󵄩󵄩󵄩 (T − z) P∂ i f n 󵄩󵄩󵄩󵄩󵄩2,D = 0 (19) for i = 1, 2, . . . , m − 2. Since T has Bishop’s property (β) [12], we have lim n→∞ 󵄩󵄩󵄩󵄩󵄩 P∂ i f n 󵄩󵄩󵄩󵄩󵄩2,D 0 = 0 (20) for i = 1, 2, . . . , m − 2, where D 0 denotes a disk with σ(T) ⫋ D 0 ⫋ D. From (18) and (20), we get that lim n→∞ 󵄩󵄩󵄩󵄩󵄩 ∂ i f n 󵄩󵄩󵄩󵄩󵄩2,D 0 = 0 (21) for i = 1, 2, . . . , m − 2. Lemma 4. Let T = ( T1 T2 T 3 T 4 ) ∈ B(H ⊕ H), where T i are mutually commuting, and both T 1 and T 4 areM-hyponormal operators. For any positive integer m and any bounded disk D in C containing σ(T), define the map V m : H ⊕ H → H(D) by


Introduction and Preliminaries
Let  be a complex separable Hilbert space and let () denote the algebra of all bounded linear operators on .If  ∈ (), we write (), (), (), and   () for the null space, the range space, the spectrum, and the approximate point spectrum of , respectively.An operator  is called Fredholm if () is closed, () := dim () < ∞, and () := dim ( * ) < ∞.The index of a Fredholm operator  is given by () = () − ().An operator  is called Weyl if it is Fredholm of index zero.The Weyl spectrum of  [1] is defined by () := { ∈ C :  −  is not Weyl}.
We consider the sets Φ + () where iso () denotes the isolated points of ().
Let  ∈ ().As an easy extension of normal operators, hyponormal operators have been studied by many mathematicians.Though there are many unsolved interesting problems for hyponormal operators (e.g., the invariant subspace problem), one of recent trends in operator theory is studying natural extensions of hyponormal operators.So we introduce some of these nonhyponormal operators.An operator  is said to be -hyponormal if there exists a real positive number  such that Evidently, There is a vast literature concerning -hyponormal operators (see [3][4][5], etc.).We also note that an operator  need not be hyponormal even though  and  * are both hyponormal.To see this, consider the operator where  is the unilateral shift on  2 and  :  2 →  2 is given by ( 1 ,  2 ,  3 , . ..) = (2 1 , 0, 0, . ..).Then a direct calculation shows that for all  ∈ C and for all  ∈  2 ⊕  2 , which says that  and  * are both -hyponormal.But since while is not hyponormal.
Let  be the coordinate in the complex plane C and let () denote the planar Lebesgue measure.Fix a complex (separable) Hilbert space  and a bounded (connected) open subset  of C. We will denote by  2 (, ) the Hilbert space of measurable functions  :  → , such that The Bergman space for  is defined by  2 (, ) = In 1984, Putinar showed in [6] that every hyponormal operator is subscalar, and then in 1987, Brown used this result to prove that a hyponormal operator with rich spectrum has a nontrivial invariant subspace (see [7]).There have been a lot of generalizations of such beautiful consequences (see [8][9][10][11]).In this paper, we provide some conditions for 2 × 2 operator matrices whose diagonal entries are -hyponormal operators to be subscalar.As a consequence, we obtain that Weyl type theorem holds for such operator matrices.

Subscalarity
Lemma 1 (see [6,Proposition 2.1]).For a bounded open disk  in the complex plane C, there is a constant   such that for an arbitrary operator  ∈ () and  ∈  2 (, ) we have where  denotes the orthogonal projection of  2 (, ) onto the Bergman space  2 (, ).

Corollary 2.
Let  be as in Lemma 1.If  ∈ () is an hyponormal operator, then there exists a constant   such that for all  ∈ C and  ∈  2 (, ) where  denotes the orthogonal projection of  2 (, ) onto the Bergman space  2 (, ).
Proof.This follows from Lemma 1 and the definition of hyponormal operator. where and 1 ⊗ ℎ denotes the constant function sending any  ∈  to ℎ ∈  ⊕ .Then the following statements hold.
Let Γ be a curve in    surrounding ().