A Note on the Range of a Derivation

. Let H be a separable inﬁnite dimensional complex Hilbert space, and let L ( H ) denote the algebra of all bounded linear operators on H into itself. Given A, B ∈ L ( H ), deﬁne the generalized derivation δ A,B ∈ L ( L ( H )) by δ A,B ( X ) = AX − XB . An operator A ∈ L ( H ) is P -symmetric if AT = TA implies AT ∗ = T ∗ A for all T ∈ C 1 ( H ) (trace class operators). In this paper, we give a generalization of P -symmetric operators. We initiate the study of the pairs ( A, B ) of operators A, B ∈ L ( H ) such that R ( δ A,B ) W ∗ = R ( δ A ∗ ,B ∗ ) W ∗ , where R ( δ A,B ) W ∗ denotes the ultraweak closure of the range of δ A,B . Such pairs of operators are called generalized P -symmetric. We establish a characterization of those pairs of operators. Related properties of P -symmetric operators are also given.


Introduction and Notation
Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators acting on H into itself. Given A, B ∈ L(H), we define the generalized derivation δ A,B : L(H) −→ L(H) by δ A,B (X) = AX − XB, we simply write δ A for δ A,A .
An operator A ∈ L(H) is called D-symmetric if R(δ A ) = R(δ A * ), where R(δ A ) denotes the norm closure of the range R(δ A ) of δ A . Clearly A is Dsymmetric if and only if R(δ A ) is a self-adjoint subspace of L(H). Examples of D-symmetric operators include the normal operators, isometries and hyponormal weighted shifts. The properties of D-symmetric operators have been considered in a number of papers (see for example [1], [5], [6], [3], [4], [7], [9], [10] and [11]).
In [2] it is proved that if A is D-symmetric, then A has the property (F P ) C1(H) , that is, AT = T A implies A * T = T A * for every T ∈ C 1 (H) (trace class operators). Operators A satisfying the property (F P ) C1(H) are termed P -symmetric.
S. Bouali and J. Charles introduced P -symmetric operators, and they gave some basic properties of this class of operators ( [5], [6]). In this paper, we study The present paper investigates also the class of P -symmetric operators. We prove that if A is a rationally cyclic subnormal operator, then A is Dsymmetric. A well-known result of S. Bouali and J. Charles [5] says that an operator A ∈ L(H) is P -symmetric if and only if R(δ A ) W * is self-adjoint. So, for a P -symmetric operator A we consider the following sets: C • (A) = {C ∈ L(H) : We establish some new properties concerning these subalgebras of L(H). We present some examples and counterexamples of P -symmetric and essentially D-symmetric operators.
We conclude this section with some notation and terminology. An operator on H will always be understood to be a bounded linear transformation from H Volumen 56, Número 2, Año 2022 into itself. The algebra of all bounded linear operators on H will be dented by L(H). Given A ∈ L(H), we shall denote the kernel, the orthogonal complement of the kernel and the closure of the range of A by ker(A), ker ⊥ (A) and R(A), respectively. The spectrum of A will be denoted by σ(A), and the restriction of A to an invariant subspace M will be denoted by A|M . A closed subspace M of H is said to reduce A if AM ⊆ M and AM ⊥ ⊆ M ⊥ , that is, if M and M ⊥ are both invariant under A. For λ ∈ C, let λ denote the complex conjugate of λ. A complex number λ is said to be a reducing eigenvalue for A if ker(A − λI) reduces A, where I is the identity operator. For vectors x and y in H we denote by x ⊗ y the rank-one operator defined by x ⊗ y(z) = z, y x for all z ∈ H.  Let B be a Banach space and let S be a subspace of B. Denote by B the set of all bounded linear functionals. We define the annihilator of S by Any other notation will be explained as and when required.

Preliminaries
If A ∈ L(H), then the following two statements are equivalent.
, the corresponding element of the Calkin algebra, is D-symmetric and (2) P (H) (the set of P -symmetric operators) is self-adjoint.
then A is not P -symmetric.
Example 2.6. Let (e n ) n∈N be an orthonormal basis for H. Let H • = ∨{e 1 , e 2 , e 3 } denote the linear subspace of H generated by the set {e 1 , e 2 , e 3 }. Let A • be defined by where I is the identity operator. It is easily verified that It follows that A is not P -symmetric.
, its corresponding element of the Calkin algebra, is D-symmetric.
The following result is an immediate consequence of Theorem 2.1. (ii) An essentially normal operator A is D-symmetric if and only if A is P -symmetric.
(iii) An operator in the trace class is P -symmetric if and only if it is normal.
Remark 2.9. Let (e n ) n∈N (respectively (e n ) n∈Z ) be an orthonormal basis for H. Let S be the unilateral (respectively bilateral) shift Se n = ω n e n+1 where n ∈ N (respectively n ∈ Z) with nonzero weights ω n . By taking a unitarily equivalent weighted shift, we may assume that ω n = |ω n | > 0.
In [6] it is shown that S is P -symmetric if and only if S satisfies the total products condition, that is, Volumen 56, Número 2, Año 2022 Example 2.10. We now present an example of an essentially D-symmetric which is not P -symmetric. We define our operator A as follows.
Let (e n ) n∈Z be an orthonormal basis for H. Set It is obvious that A is essentially normal. Then it follows from Theorem 2.1 in [1] ( which is valid in any C * -algebra), that A is essentially D-symmetric. However the weights of A don't satisfy the total products condition, and so A is not P -symmetric.
Since the weights of T satisfy the total products condition, then T is Psymmetric. It follows from Lemma 2 in [11] that T is not essentially D-symmetric. On the other hand, we have that T 2 is unitary, and hence T 2 is D-symmetric. But T is not D-symmetric.

Main Results
The proof of the preceding lemma is the same as the proof of Theorem 3 [12]. Proof. Observe that the assertion R(δ A,B ) We get from lemma 3.1 that Hence, it follows that This completes the proof. Proof. We must show that R(δ A,B ) for every T ∈ C 1 (H).
It suffices to exhibit a trace class operator T for which f T ∈ R(δ A,B ) • but f T ∈ R(δ A * ,B * ) • . Let us define the rank one operator T = x ⊗ y.
Then for any Y ∈ L(H) we have Define an operator X ∈ L(H) by X = y ⊗ (B − λ) * x. Then, it follows that which completes the proof.  Proof. Assume that A and B are P -symmetric operators with disjoint spectra.
, then there exists a sequence (X α ) α of elements in L(H) such that (AX α − X α B) α converges in the ultra-weak topology (or the weak * operator topology) to T , in symbols we write On H ⊕ H consider the operators L, S and Y α defined as Hence, we get S ∈ R(δ L ) W * . Since A and B are P -symmetric with disjoint spectra, then we obtain from Theorem 2.6 in [3], that L is P -symmetric. Thus, there exists a sequence (Z α ) α in L(H ⊕ H) for which δ L * (Z α ) W * −→ S. A simple calculation shows that there exists a sequence (U α ) α in L(H) such that The argument to verify the reverse inclusion is identical to the above and thus the proof is complete.  If (A, B) is P -symmetric, then Proof. Suppose that (A, B) is P -symmetric. Then we have If X is any operator in L(H) then From this it follows that By the same argument as above, we prove that . So the proof is complete.   where Rat(σ(A)) is the set of rational functions with poles off σ(A), and e is called a rationally cyclic vector for A.
Theorem 3.12. Let A ∈ L(H). If A is a rationally cyclic subnormal operator, then A is D-symmetric.
Proof. Suppose that A is a rationally cyclic subnormal operator, and let T ∈ C 1 (H) be such that AT = T A. It follows from Yoshino's result [14] that T is subnormal. It is well-known that any compact subnormal operator is normal. We get that T is normal . Hence, it follows from Fuglede-Putnam theorem that AT = T A implies AT * = T * A. Thus, A is P -symmetric.
Since A is a rationally cyclic subnormal operator, it results from Shaw and Berger's Theorem [2] that [A] is normal. Then A is essentially D-symmetric. This proves that A is D-symmetric. Remark 3.14. It is known that A is P -symmetric if and only if R(δ A ) W * is a self-adjoint subspace of L(H). Hence, for a P -symmetric operator A, it is natural to introduce the following subalgebras: It is well-known that if H is of finite dimension ( [13]), then Theorem 3.15. Let A ∈ L(H) be a P -symmetric operator. Then the following statements are equivalent: (ii) A has no reducing eigenvalues. . Then there exist a non-zero T in the trace class such that f T vanishes on R(δ A ), that is, AT = T A. Since A is P -symmetric, then AT = T A and T ∈ C 1 (H) implies AT * = T * A. It follows that A(T + T * ) = (T + T * )A and A(T − T * ) = (T − T * )A. Hence A commutes with a non-zero trace class operator. Consequently, A has a finite dimensional reducing subspace H • . Clearly, A|H • is P -symmetric, and so A|H • is normal by Corollary 2.1 (iii). Thus T has a reducing eigenvalue.
We include the following properties for the sake of completeness. We omit the proofs which are based entirely on those of C. Gupta and P. Ramanujan for D-symmetric operators [8].
Corollary 3.16. Let A ∈ L(H) be a P -symmetric operator with C • (A) = K(H). Then we have the following assertions: (i) A is essentially normal.
(ii) A has no reducing eigenvalue.