Decomposable Operators Research Articles

Local spectral properties of commutators

For a pair of continuous linear operatorsTandSon complex Banach spacesXandY, respectively, this paper studies the local spectral properties of the commutatorC(S, T) given byC(S, T)(A): =SA−ATfor allA∈L(X, Y). Under suitable conditions onTandS, the main results provide the single valued extension property, a description of the local spectrum, and a characterization of the spectral subspaces ofC(S, T), which encompasses the closedness of these subspaces. The strongest results are obtained for quotients and restrictions of decomposable operators. The theory is based on the recent characterization of such operators by Albrecht and Eschmeier and extends the classical results for decomposable operators due to Colojoară, Foiaş, and Vasilescu to considerably larger classes of operators. Counterexamples from the theory of semishifts are included to illustrate that the assumptions are appropriate. Finally, it is shown that the commutator of two super-decomposable operators is decomposable.

Functional calculi and decomposability of unbounded multiplier operators in LP(ℝN)

It is known, for each 1<p<∞, p≠2, that there exist differential operators in LP(ℝN) which are not (unbounded) decomposable operators in the sense of C. Foiaş. In this note we exhibit large classes of differential (and unbounded multiplier operators which are decomposable in LP(ℝN) and hence have good spectral mapping properties; the arguments are based on the existence of a sufficiently rich functional calculus. The basic idea is to take advantage of existing classical results on p-multipliers and use them to generate appropriate functional calculi.

Power regular operators

We show that for a wide class of operators T T on a Banach space, including the class of decomposable operators, the sequence { ‖ T n x ‖ 1 / n } n = 1 ∞ \left \{ {{{\left \| {{T^n}x} \right \|}^{1/n}}} \right \}_{n = 1}^\infty converges for every x x in the space to the spectral radius of the restriction of T T to the subspace ∨ n = 0 ∞ { T n x } \vee _{n = 0}^\infty \{ {T^n}x\} .

Products of Decomposable Positive Operators

AbstractIn recent years there has been a growing interest in problems of factorization for bounded linear operators. We first show that many of these problems properly belong to the category of C*-algebras. With this interpretation, it becomes evident that the problem is fundamental both to the structure of operator algebras and the elements therein. In this paper we consider the direct integral algebra with separable and infinite dimensional. We generalize a theorem of Wu (1988) and characterize those decomposable operators which are products of non-negative decomposable operators. We do this by first showing that various results on operator ranges may be generalized to “measurable fields of operator ranges”.

Scott Brown's techniques for perturbations of decomposable operators

Using Scott Brown's techniques, J. Eschmeier and B. Prunaru showed that if T is the restriction of a decomposable (or S-decomposable) operator B to an invariant subspace such that σ(T) is dominating in C/S for some closed set S, then T has an invariant subspace. In the present paper we prove various invariant subspace theorems by weakening the decomposability condition on B and strengthening the thickness condition on σ(T).

Commutative Banach algebras and decomposable operators

For a multiplication operator on a semi-simple commutative Banach algebra, it is shown that the decomposability in the sense of Foias is equivalent to weak and to super-decomposability. Moreover, it can also be characterized by a convenient continuity condition for the Gelfand transform on the spectrum of the underlying Banach algebra. This result implies various permanence properties for decomposable multiplication operators and leads also to a useful characterization of the regularity for a semi-simple commutative Banach algebra. Finally, the greatest regular closed subalgebra of a commutative Banach algebra is investigated, and some applications to decomposable convolution operators on locally compact abelian groups are given.

Open-restriction decomposition property

Let T be a closed linear operator on a complex Banach space X. If T has the spectral decomposition property (SDP) and its restriction to each spectral manifold X( T, [ G ∩ σ( T)] −), G open, also has SDP, then the authors say that T has the open-restriction decomposition property (ORDP). Two criteria for this property are given. It is also proved that Albrecht's example of a decomposable operator that is not strongly decomposable nevertheless has ORDP.

Universal notions characterizing spectral decompositions

In this note we characterize certain types of spectral decomposition in terms of “universal” notions valid for any operator on a Banach space. To be precise, let X be a complex Banach space and let T be a bounded linear operator on X. If F is a closed set in the plane C, let X(T, F) consist of all y ∈ X satisfying thes identitywhere f:C\F → X is analytic. It is then easy to see that X(T, F) is a T-invariant linear manifold in X. Moreover, if y ∈ X thenis a compact subset of the spectrum σ(T). Our aim is to give necessary and sufficient conditions for a decomposable or strongly decomposable operator in terms of X(T, F) and γ(y, T). Recall that T is decomposable if whenever G1G2 are open and cover C there exist T-invariant closed linear manifolds M1, M2 with X= M1 + M2 and σ(T | M1) ⊂ Gi(i = 1,2) (equivalently, σ(T | Mi)⊂ Ḡi, see [4, p. 57]). In this case, X(T, F) is norm closed if Fis closed and each y in X has a unique maximally defined local resolvent satisfying (1.1) on C\Fy; Fy is called the local spectrum σ(y, T) and coincides with γ(y, T). Hence T has the single valued extension property (SVEP); i.e., zero is the only analytic function f:V → X satisfying (z − T)f(z) = 0 on V. If T is decomposable and the restriction T | X(T, F) is also decomposable for each closed F, then T is called strongly decomposable. We point out that Albrecht [2] has shown by example that not every decomposable operator is strongly decomposable, while Eschmeier [6]has given a simpler construction to show that this phenomenon occurs even in Hilbert space.

Automatic continuity of intertwining linear operators on Banach spaces

We extend the automatic continuity theory for linear operators θ:X→Y which intertwine two given bounded linear operatorsT∈LX andS∈LY on Banach spacesX andY, respectively. This is done both by relaxing the intertwining conditionSθ=θT and by enlarging the classes of operatorsT, resp.S, well beyond the decomposable operators. Among the operatorsS captured by these extensions are multipliers on commutative semi-simple Banach algebras.

The Algebra of Decomposable Operators in Direct Integrals of Not Necessarily Separable Hilbert Spaces

Considering direct integrals of not necessarily separable Hilbert spaces we examine the question whether the algebra of decomposable operators is the commutant of the algebra of diagonalizable operators. Using the continuum-hypothesis we prove this relation, if the set of square integrable vector fields is generated by a subset ${\Gamma _0}$ such that $|{\Gamma _0}| \leq |{\mathbf {R}}|$. For the general case, a counterexample is given.

Semismoothness and decomposition of maximal normal operators

A function is semismooth if and only if its subdifferential is semicontinuous. All the semicontinuous operators form a big linear subspace of the maximal normal operator space. By “big,” we mean that this subspace includes all the continuous operators and all the maximal monotone operators. All the decomposable maximal normal operators form another big subspace.

Commuting boundedly decomposable perturbations

The author proves that perturbation of a decomposable operator by a commuting boundedly decomposable operator is decomposable, thus generalizing a theorem of C. Apostol. It is also proved that the sum of a regular, A-spectral operator and a commuting boundedly decomposable operator is strongly decomposable, while the sum of two commuting boundedly decomposable operators is generalized spectral. Examples show that boundedly decomposable operators form a proper intermediate class between spectral and generalized spectral operators; thus the generalization of Apostol's theorem is not trivial.

The algebra of decomposable operators in direct integrals of not necessarily separable Hilbert spaces

Considering direct integrals of not necessarily separable Hilbert spaces we examine the question whether the algebra of decomposable operators is the commutant of the algebra of diagonalizable operators. Using the continuum-hypothesis we prove this relation, if the set of square integrable vector fields is generated by a subset Γ 0 {\Gamma _0} such that | Γ 0 | ≤ | R | |{\Gamma _0}| \leq |{\mathbf {R}}| . For the general case, a counterexample is given.

Nonanalytic functional calculi and spectral maximal spaces

We are concerned with the algebraic representation of the spectral maximal spaces for certain classes of decomposable operators on Banach spaces. The main emphasis will be on those operators which admit a functional calculus on a suitable algebra of functions such as, for instance, differentiable functions, Lipschitz functions, or functions with absolutely convergent Fourier series. Using appropriate partitions of unity, we shall give a unified approach to various previous results in this area as well as to certain extensions and variants thereof. In the case of spectral operators, we shall obtain the optimal results without any assumption on the underlying Banach space or the Boolean algebra of projections corresponding to the spectral measure.

Duality of closed $S$-decomposable operators

Duality of closed $S$-decomposable operators

Arens regularity ofK(X)

LetX be a reflexive Banach space andK(X) the operator algebra of compact linear operatorsu:X→X. In this note we prove the following two results: a) Any decomposable bilinear operator fromK(X)×K(X) intoK(X) is Arens regular; b) ifX has an unconditional basis then any bounded bilinear operator fromK(X)×K(X) intoK(X) is Arens regular.

Strongly analytic spaces in spectral decomposition

It is now well-known that decomposable operators have a rich structure theory; in particular, an operator is decomposable iff its adjoint is [3]. There are many other criteria for decomposability [8], [9]. In Theorem 2.2 of this paper (see below) we give several new ones. Some of these (e.g. (ii), (iii)) are “relaxations” of conditions given in [7] and [8]. Assertion (vi) is a version of a result in [10]. Characterizations (iv)and (v) are novel in two respects. For instance, (v) states that an operator Tcan be “patched” together into a decomposable operator if it has an invariant subspace Y such that T | Y and the coinduced operator T | Y are both decomposable. Secondly, in this way the strongly analytic subspace appears in the theory of spectral decomposition.

A decomposable Hilbert space operator which is not strongly decomposable

In 1978 E. Albrecht constructed a Banach space operator which is decomposable in the sense of C. Foias, but not strongly decomposable. In the present note we describe a method which allows to construct examples of this kind in a systematic and much simpler way. In particular, we exhibit a decomposable, but not strongly decomposable operator on a Hilbert space and thus answer a corresponding question of M.Radjabalipour.

New criteria for a decomposable operator

New criteria for a decomposable operator

Spectral Cutting for a Class of Subnormal Operators

Let $S$ be a subnormal decomposable operator on a Hilbert space $\mathcal {H}$. (The dual of the Bergman shift belongs to this class.) It is shown that for any closed set $\delta$ with nonempty intersection with the spectrum of $S,\mathcal {H}$ can be decomposed as $M(\delta ) \oplus \overline {M(\delta ’)} \oplus {M^ * }$, where $M(\delta )$ and $\overline {M(\delta ’)}$ are hyperinvariant under $T$, and ${M^ * }$ is invariant under ${T^ * }$ with spectrum of ${T^ * }|{M^ * }$ contained in the conjugate of the boundary of $\delta$. The minimal normal extension of the subnormal operator $T|M(\delta )$ is also considered.