Conjugate Linear Operator Research Articles

Partial stability of linear systems with respect to a given component of the phase vector

A new geometric approach to the study of the partial stability of linear systems is proposed, which is based on the application of the geometric theory of linear operators. Using the theory of conjugate spaces and conjugate linear operators, bases are constructed in which the system under study takes the canonical form. A cyclic subspace with respect to the conjugate linear operator is considered. A basis is constructed for the dual space of a linear operator, in which its matrix takes the canonical form. This basis corresponds to the dual basis of the original linear space. Then, in a pair of bases of dual spaces the system under study takes the simplest form. The geometric properties of the system are realized using a non-singular linear transformation in the space of a part of the components of the system’s phase vector. This allows us to decompose the system under study in order to obtain necessary and sufficient conditions for the partial stability of the linear system. In an equivalent system, an independent subsystem is distinguished, whose nature of stability determines the behavior of the investigated component of the original system’s phase vector. The relationship between the partial stability of the system and the existence of an invariant subspace of a linear operator characterizing the dynamics of the system is established. The canonical form of the resulting subsystem makes it easy to exclude auxiliary variables and write an equation equivalent to this system. The application of the obtained results to the solution of the problem of partial stability for linear systems with constant coefficients belonging to the classes of ordinary differential equations, discrete systems and systems with deviating argument is shown. An example of a linear system of differential equations is given to illustrate the result obtained.

Nonlinear preservers of group invertible operators

Let [Formula: see text] be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space [Formula: see text]. We prove that a bijective bicontinuous map [Formula: see text] on [Formula: see text] preserves the difference of group invertible operators in both directions if and only if [Formula: see text] is either of the form [Formula: see text] or of the form [Formula: see text], where [Formula: see text] is a nonzero scalar, [Formula: see text] and [Formula: see text] are two bounded invertible linear or conjugate linear operators.

Additive preservers of Drazin invertible operators with bounded index

Let B(X) be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space X. Given an integer n ≥ 1, we show that an additive surjective map Φ on B(X) preserves Drazin invertible operators of index non-greater than n in both directions if and only if Φ is either of the form Φ(T) = αATA −1 or of the form Φ(T) = αBT *B −1 where α is a non-zero scalar, A: X → X and B: X* → X are two bounded invertible linear or conjugate linear operators.

Lie ring isomorphisms between nest algebras on Banach spaces

Let N and M be nests on Banach spaces X and Y over the (real or complex) field F and let AlgN and AlgM be the associated nest algebras, respectively. It is shown that a map Φ:AlgN→AlgM is a Lie ring isomorphism (i.e., Φ is additive, Lie multiplicative and bijective) if and only if Φ has the form Φ(A)=TAT−1+h(A)I for all A∈AlgN or Φ(A)=−TA⁎T−1+h(A)I for all A∈AlgN, where h is an additive functional vanishing on all commutators and T is an invertible bounded linear or conjugate linear operator when dimX=∞; T is a bijective τ-linear transformation for some field automorphism τ of F when dimX<∞.

Maps preserving the harmonic mean or the parallel sum of positive operators

Let H be a complex Hilbert space. The symbol A ! B stands for the harmonic mean of the positive bounded linear operators A , B on H in the sense of Ando. In this paper we describe the general form of all automorphisms of the set of positive operators with respect to that operation. We prove that any such transformation is implemented by an invertible bounded linear or conjugate-linear operator on H . Similar results concerning the parallel sum and the arithmetic mean in the place of the harmonic mean are also presented.

Maps on positive operators preserving Lebesgue decompositions

Let H be a complex Hilbert space. Denote by B(H)+ the set of all positive bounded&nbsp;linear operators on H. A bijective map&nbsp;∅: B(H)+ → B(H)+ is said to preserve Lebesgue decompositions&nbsp;in both directions if for any quadruple A,B,C,D of positive operators, B = C + D is an&nbsp;A-Lebesgue decomposition of B if and only if&nbsp;φ(B) =&nbsp;φ(C) +&nbsp;φ(D) is a&nbsp;φ(A)-Lebesgue decomposition&nbsp;of φ(B). It is proved that every such transformation&nbsp;φ&nbsp;is of the form φ(A) = SAS∗ (A ∈ B(H)+)&nbsp;for some invertible bounded linear or conjugate-linear operator S on H.

Maps preserving the geometric mean of positive operators

Let H H be a complex Hilbert space. The symbol A # B A\# B stands for the geometric mean of the positive bounded linear operators A , B A,B on H H in the sense of Ando. In this paper we describe the general form of all automorphisms of the set of positive operators with respect to the operation # \# . We prove that if dim ⁡ H ≥ 2 \dim H\geq 2 , any such transformation is implemented by an invertible bounded linear or conjugate-linear operator on H H .

Additive maps preserving similarity or asymptotic similarity on

Let H be an infinite-dimensional complex separable Hilbert space and denote the algebra of all bounded linear operators acting on H. We show that an additive continuous surjective map Φ on is asymptotic similarity preserving if and only if it is similarity preserving, and in turn, if and only if there exist a scalar c and an invertible bounded linear or conjugate linear operator A on H such that either Φ(T)=cATA −1 for all T or Φ (T)=cAT*A −1 for all T.

Maps preserving the nilpotency of products of operators

Let B ( X ) be the algebra of all bounded linear operators on the Banach space X, and let N ( X ) be the set of nilpotent operators in B ( X ) . Suppose ϕ : B ( X ) → B ( X ) is a surjective map such that A , B ∈ B ( X ) satisfy AB ∈ N ( X ) if and only if ϕ ( A ) ϕ ( B ) ∈ N ( X ) . If X is infinite dimensional, then there exists a map f : B ( X ) → C ⧹ { 0 } such that one of the following holds: (a) There is a bijective bounded linear or conjugate-linear operator S : X → X such that ϕ has the form A ↦ S [ f ( A ) A ] S - 1 . (b) The space X is reflexive, and there exists a bijective bounded linear or conjugate-linear operator S : X′ → X such that ϕ has the form A ↦ S[ f( A) A′] S −1. If X has dimension n with 3 ⩽ n < ∞, and B ( X ) is identified with the algebra M n of n × n complex matrices, then there exist a map f : M n → C ⧹ { 0 } , a field automorphism ξ : C → C , and an invertible S ∈ M n such that ϕ has one of the following forms: A = [ a ij ] ↦ f ( A ) S [ ξ ( a ij ) ] S - 1 or A = [ a ij ] ↦ f ( A ) S [ ξ ( a ij ) ] t S - 1 , where A t denotes the transpose of A. The results are extended to the product of more than two operators and to other types of products on B ( X ) including the Jordan triple product A ∗ B = ABA. Furthermore, the results in the finite dimensional case are used to characterize surjective maps on matrices preserving the spectral radius of products of matrices.

Elementary operators on self-adjoint operators

Let H be a Hilbert space and let A and B be standard ∗-operator algebras on H. Denote by A s and B s the set of all self-adjoint operators in A and B , respectively. Assume that M : A s → B s and M ∗ : B s → A s are surjective maps such that M ( A M ∗ ( B ) A ) = M ( A ) B M ( A ) and M ∗ ( B M ( A ) B ) = M ∗ ( B ) A M ∗ ( B ) for every pair A ∈ A s , B ∈ B s . Then there exist an invertible bounded linear or conjugate-linear operator T : H → H and a constant c ∈ { − 1 , 1 } such that M ( A ) = c T A T ∗ , A ∈ A s , and M ∗ ( B ) = c T ∗ B T , B ∈ B s .

Additive Maps Preserving Similarity of Operators on Banach Spaces

Let X be an infinite–-dimensional complex Banach space and denote by ℬ(X) the algebra of all bounded linear operators acting on X. It is shown that a surjective additive map Φ from ℬ(X) onto itself preserves similarity in both directions if and only if there exist a scalar c, a bounded invertible linear or conjugate linear operator A and a similarity invariant additive functional φ on ℬ(X) such that either Φ(T) = cATA−1 + φ(T)I for all T, or Φ(T) = cAT*A−1 + φ(T)I for all T. In the case where X has infinite multiplicity, in particular, when X is an infinite–dimensional Hilbert space, the above similarity invariant additive functional φ is always zero.

Additive Maps Preserving Nilpotent Operators or Spectral Radius

Let X be a (real or complex) Banach space with dimension greater than 2 and let ℬ0(X) be the subspace of ℬ(X) spanned by all nilpotent operators on X. We get a complete classification of surjective additive maps Φ on ℬ0(X) which preserve nilpotent operators in both directions. In particular, if X is infinite–dimensional, we prove that Φ has the form either Φ(T) = cATA−1 or Φ(T) = cAT'A−1, where A is an invertible bounded linear or conjugate linear operator, c is a scalar, T' denotes the adjoint of T. As an application of these results, we show that every additive surjective map on ℬ(X) preserving spectral radius has a similar form to the above with |c| = 1.

Jordan zero-product preserving additive maps on operator algebras

Let Φ : A → B be an additive surjective map between some operator algebras such that A B + B A = 0 implies Φ ( A ) Φ ( B ) + Φ ( B ) Φ ( A ) = 0 . We show that, under some mild conditions, Φ is a Jordan homomorphism multiplied by a central element. Such operator algebras include von Neumann algebras, C ∗ -algebras and standard operator algebras, etc. Particularly, if H and K are infinite-dimensional (real or complex) Hilbert spaces and A = B ( H ) and B = B ( K ) , then there exists a nonzero scalar c and an invertible linear or conjugate-linear operator U : H → K such that either Φ ( A ) = c U A U −1 for all A ∈ B ( H ) , or Φ ( A ) = c U A ∗ U −1 for all A ∈ B ( H ) .

Maps on B(H) Preserving Idempotents

Let B(H) be the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space H and the set of all idempotents in B(H). Suppose is a surjective map satisfying if and only if . Then ϕ is either of the form and T is a continuous invertible linear operator on H, or of the form and T is a continuous invertible conjugate linear operator on H.

Asymptotic similarity preserving additive maps on [formula omitted

Let X be an infinite dimensional complex Banach space and denote B(X) the algebra of all bounded linear operators acting on X. We show that an additive surjective map on B(X) preserves asymptotic similarity in both directions if and only if there exist a nonzero scalar c, an invertible bounded linear or conjugate linear operator A and an asymptotic similarity invariant additive functional φ on B(X) such that either Φ( T)= cATA −1+ φ( T) I for all T or Φ(T)=cAT ∗A −1+φ(T)I for all T. In the case that X has infinite multiplicity, especially if X is an infinite dimensional Hilbert space, above asymptotic similarity invariant additive functional φ is always zero.

On the norm of symmetrised two-sided multiplications

The authors provide precise lower bounds for the completely bounded norm of the operatorTa,b: ß(H) → ß(H) defined byTa,b(x) =axb+bxaand the injective norm of the corresponding tensor. Further, they compute the norm of the operatorx↦a*xb+b*xaacting on the space of all conjugate-linear operators onH.

Classification and properties of the spectrum of additive operators

For the algebra of operators that are additive and continuous on all of a Hilbert space we introduce a classification of the spectrum corresponding to the usual one in the case of the algebra of linear operators. We study properties of the components of the spectrum. We study the properties of the spectrum of normal self-adjoint and unitary conjugate-linear operators. The classification introduced here can be modified for closed densely defined additive operators.

Automorphisms of the Poset of Skew Projections

Any automorphism of the poset B of all idempotents of the algebra B( H) of all bounded linear operators in a Hilbert space H, dim H ≥ 3 is shown to be of the form P ↦ APA −1 or P ↦ AP* A −1 with A ∈ C , where C is the set of all semi-linear bijections A: H→ H if H is finite-dimensional and the set of all continuous invertible linear or conjugate linear operators in H if H is infinite-dimensional. Further applications of the method are outlined.

On 𝐽-selfadjoint extensions of 𝐽-symmetric operators

A short proof is given (via the theory of conjugate-linear operators) of the fact that every J-symmetric operator in a Hilbert space K \mathcal {K} has a J-selfadjoint extension in K \mathcal {K} .