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On the Decomposition of Families of Quasinormal Operators Opuscula Math. 33, no. 3 (2013), 419–438 http://dx.doi.org/10.7494/OpMath.2013.33.3.419 Opuscula Mathematica ON THE DECOMPOSITION OF FAMILIES OF QUASINORMAL OPERATORS Zbigniew Burdak Communicated by Aurelian Gheondea Abstract. The canonical injective decomposition of a jointly quasinormal family of operators is given. Relations between the decomposition of a quasinormal operator T and the decompo- sition of a partial isometry in the polar decomposition of T are described. The decomposition of pairs of commuting quasinormal partial isometries and its applications to pairs of com- muting quasinormal operators is shown. Examples are given. Keywords: multiple canonical decomposition, quasinormal operators, partial isometry. Mathematics Subject Classification: 47B20, 47A45. 1. INTRODUCTION Let L(H) be the algebra of all bounded linear operators on a complex Hilbert space H. If T L(H), then T ∗ stands for the adjoint of T. By (T ) and (T ) we denote the ∈ N R kernel and the range space of T respectively. By a subspace we always understand a closed subspace. The orthogonal comple- ment of a subspace H H is denoted by H⊥ or H H . The commutant of T L(H) 0 ⊂ 0 0 ∈ denoted by T 0 is the algebra of all operators commuting with T. A subspace H H 0 ⊂ is T hyperinvariant, when it is invariant for every S T 0. An orthogonal projection ∈ onto H is denoted by P . A subspace H reduces operator T L(H) (or is reduc- 0 H0 0 ∈ ing for T ) if and only if P T 0. An operator is called completely non unitary (non H0 ∈ normal, non isometric etc.) if there is no non trivial subspace reducing it to a unitary operator (normal operator, isometry). Such an operator is also called pure. Let denote some property of an operator T L(H) (like being unitary, normal, W ∈ isometry etc.) If there is a decomposition H = H H such that H ,H reduce 1 ⊕ 2 1 2 operator T and T has the property while T is pure, then |H1 W |H2 T = T T |H1 ⊕ |H2 c AGH University of Science and Technology Press, Krakow 2013 419 420 Zbigniew Burdak is called a canonical decomposition of T. The classical example of a canonical W decomposition is the Wold decomposition [11]. Wold showed that any isometry can be decomposed into a unitary and a completely non unitary operators. An isometry n S L(H) is called a unilateral shift if H = S ( (S∗)). A completely non ∈ n 0 N unitary isometry is a unilateral shift. ≥ L An operator T L(H) is called quasinormal, if it is normal on (T ) (i.e. ∈ R TT ∗T = T ∗TT ). The class of quasinormal operators has been introduced by Brown in [2]. Quasinormal operators are subnormal. Every quasinormal operator T can be decomposed to N S A, where N is normal, S is a unilateral shift and A is a positive, ⊕ ⊗ injective operator. For a quasinormal operator T holds (T ) (T ∗). Thus (T ) N ⊂ N N reduces a quasinormal operator. Recall that T = √T T. An operator W is a partial | | ∗ isometry, if the restriction W (W ) is an isometry. Every T L(H) has the polar |N ⊥ ∈ decomposition T = W T , where W is a partial isometry. Operator T is quasinormal | | if and only if W T = T W. Recall after [7] that a family of operators L(H) is | | | | A ⊂ called jointly quasinormal, if ST ∗T = T ∗TS holds for all S, T . ∈ A Assume that, for some property , there are canonical decompositions of W W operators T ,T L(H). If there is a decomposition H = H H H H , 1 2 ∈ 11 ⊕ 12 ⊕ 21 ⊕ 22 where each H reduces operators T ,T for i, j = 1, 2, such that: T has the ij 1 2 1|Hij property for i = 1, is pure for i = 2 and T has the property for j = 1 and W 2|Hij W is pure for j = 2, then T = T T T T for k = 1, 2 k k|H11 ⊕ k|H12 ⊕ k|H21 ⊕ k|H22 is called a multiple canonical decomposition. Multiple canonical decompositions W can be defined in a similar way for families of operators. Recall that operators T ,T L(H) doubly commute if T ,T ∗ T 0. Recall after [4] that a family of doubly 1 2 ∈ 1 1 ∈ 2 commuting operators has a multiple canonical decomposition if each operator in the family has a (single) canonical decomposition. However, the doubly commutativity assumption is rather strong. Only a normal operator can doubly commute with itself. On the other hand, a pair T,T has a multiple canonical decomposition if T has a canonical decomposition. In the present paper we show that, although there does not need to exist a normal canonical decomposition of a jointly quasinormal family of operators, there is an injective canonical decomposition. We also give a generalization of this decomposition to a pair of commuting quasinormal partial isometries. The generalization is not a canonical decomposition. In the last paragraph we give some applications of this decomposition to arbitrary pairs of commuting quasinormal operators. 2. DECOMPOSITIONS OF A QUASINORMAL OPERATOR By the model given by Brown a quasinormal operator has a normal canonical de- composition. Let T L(H) be a quasinormal operator. Recall after [10] that if T is ∈ additionally a contraction, then the subspace (I T ∗T ) is the maximal subspace N − reducing T to an isometry. Note that a subspace reduces operator T if and only if it reduces αT, for α C 0 . If α = 1, then (I T ∗T ) = (I (αT )∗(αT )) . Since αT ∈ \{ } | | 6 N − 6 N − On the decomposition of families of quasinormal operators 421 is not a contraction for sufficiently large α , it is not known whether (I (αT )∗(αT )) | | N − reduces T. In this section the existence of the maximal subspace reducing a bounded quasinormal operator to an isometry will be proved. As a consequence, any eigenspace of T reduces operator T to an isometry weighted by the corresponding eigenvalue. | | Moreover, such a subspace is T hyperinvariant. | | Remark 2.1. Let A L(H) be a positive operator. For x (I A) we have ∈ ∈ N − x = Ax (A). Consequently, the subspace (I A) is orthogonal to (A). An ∈ R N − N arbitrary x (A A2) can be decomposed to x = (x Ax)+Ax (A) (I A). ∈ N − − ∈ N ⊕N − It follows that (A A2) (I A) (A). Since the reverse inclusion is obvious, N − ⊂ N − ⊕ N we have (A A2) = (I A) (A). N − N − ⊕ N For arbitrary x (I A2), we have ∈ N − 0 (A(x Ax), x Ax) = (Ax x, x Ax) = x Ax 2 0. ≤ − − − − −k − k ≤ Consequently, (I A2) = (I A). N − N − A decomposition of an operator T is called T hyperinvariant, when subspaces in the corresponding decomposition of the Hilbert space H = i Hi are T hyperinvariant. It follows that Hi and H Hi are T hyperinvariant and consequently subspaces Hi L reduce every operator in T 0. Proposition 2.2. Let T L(H) and T = √T T . Then the decomposition ∈ | | ∗ H = ( T T T ) (I T ) (T ) R | | − ∗ ⊕ N − | | ⊕ N is T hyperinvariant. | | Proof. Since the operator T T ∗T is self-adjoint, then | | − H = ( T T T ) ( T T ∗T ). R | | − ∗ ⊕ N | | − By Remark 2.1, we obtain the decomposition ( T T ∗T ) = ( T ) (I T ). N | | − N | | ⊕ N − | | Obviously commutants of T and I T are equal. Since the kernel of an operator is a | | −| | hyperinvariant subspace, then (I T ) and ( T ) = (T ) are T hyperinvariant. N − | | N | | N | | The orthogonal complement of a subspace, which is hyperinvariant for a self-adjoint operator is also a hyperinvariant subspace. Thus ( T T T ) is T hyperinvariant R | | − ∗ | | as well. As a corollary we obtain the following decomposition. Proposition 2.3. Let T L(H) be a quasinormal operator. There is a decomposition ∈ 2 2 H = (T T T T ) (I T ∗T ) (T ), R ∗ − ∗ ⊕ N − ⊕ N where the subspaces are the maximal reducing operator T, such that: (i) T (T T T 2T 2) is a completely non isometric, injective, quasinormal operator, |R ∗ − ∗ 422 Zbigniew Burdak (ii) T (I T T ) is an isometry, |N − ∗ (iii) T (T ) = 0. |N Moreover, each of these subspaces is T hyperinvariant. | | Proof. Applying Proposition 2.2 to the operator T ∗T we obtain the decomposition 2 H = (T T (T T ) ) (I T ∗T ) (T ∗T ). R ∗ − ∗ ⊕ N − ⊕ N Since T ∗T = T ∗T , the decomposition is T ∗T hyperinvariant. Commutants of T ∗T | | and its square root T are equal. Thus the decomposition is also T hyperinvariant. | | 2 2 2 | | Since (T ) = (T ∗T ) and by quasinormality (T ∗T ) = T ∗ T the decomposition is N N equivalent to 2 2 H = (T T T T ) (I T ∗T ) (T ). R ∗ − ∗ ⊕ N − ⊕ N Since T is quasinormal, it commutes with T . Consequently, subspaces obtained in | | the decomposition reduce T. Since for an isometry T ∗T = I, then a subspace reducing T, reduces the oper- ator to an isometry if and only if it is a subspace of (I T ∗T ). On the other N − hand we have shown that (I T ∗T ) reduces T. Thus it is the maximal subspace N − reducing T to an isometry. Since (T ) reduces T, it is the maximal subspace re- N ducing T to a zero operator. Since (T T T 2T 2) is the orthogonal complement R ∗ − ∗ of (T ) (I T ∗T ), it is the maximal subspace reducing T to a completely non N ⊕ N − isometric, injective, quasinormal operator.
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