Some properties of concave operators

We present some 2-isometric lifting and extension results for Hilbert space concave operators. For a special class of concave operators we study their Cauchy dual operators and discuss conditions under which these operators are subnormal. In particular, the quasinormality of compressions of such operators is studied. Introduction and preliminariesPreamble. Extensions and liftings are classical notions in Operator Theory. To give some examples, we recall that a linear bounded Hilbert space operator is an isometry if and only if it is the restriction of a unitary operator to an invariant subspace. Also, it is known from the Sz.-Nagy-Foias dilation theory that an operator C is a contraction if and only if it lifts to an isometry V ; that is if and only if its adjoint C * is the restriction of a coisometry V * to an invariant subspace (see [14,27]).In this paper, we prove some 2-isometric lifting and extension results for Hilbert space concave operators, that is for operators satisfying the inequality (1.1) below. A 2-isometry is an operator for which the equality in (1.1) holds true. The notion of Cauchy dual operator for a left invertible operator is more recent, being introduced in 2001 by Shimorin in his seminal study [24] of Wold-type decompositions and wandering subspaces. Here we study the Cauchy dual operators for the special class of concave operators satisfing the condition (1.3) below. Notation and basic definitions. For a complex Hilbert spaces H we denote by B(H) the Banach algebra of all bounded linear operators on H with the unit element I = I H (the identity operator). For T ∈ B(H) the kernel and the (closed) range of T are denoted by N (T ) respectively R(T ). Also, T * ∈ B(H) stands for the adjoint operator of T , and the orthogonal projection in B(H) onto a closed subspace M ⊂ H is denoted by P M . For T ∈ B(H) we consider the operator ∆ T := T * T − I. The operator T is called expansive (respectively contractive) if ∆ T ≥ 0 (respectively ∆ T ≤ 0). If T is a contraction, then D T = −∆ T is the defect operator and D T = R(D T ) is the defect space of T .

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“…Recall ( [24], [10,11], [16]) that an operator T on H is called concave if it satisfies the inequality (1.1)…”

Section: Introductionmentioning

confidence: 99%

“…Such general classes of operators were studied by many authors, from several points of view. We refer the reader to [1], [2,3], [4], [5], [6], [10,11], [12], [13], [15], [16], [18,19], [20], [21], [22], [23], [24], [25,26] for some of these contributions.…”

Section: Introductionmentioning

confidence: 99%