When products of selfadjoints are normal

Abstract

Abstract. Suppose that h, k ∈ L(H) are two selfadjoint bounded operators on a Hilbert space H. It is elementary to show that hk is selfadjoint precisely when hk = kh. We answer the following question: Under what circumstances must hk be selfadjoint given that it is normal?Let h, k ∈ L(H) be two selfadjoint bounded linear operators on a Hilbert space, shows that hk need not be selfadjoint; note that hereHowever, if either h or k is positive, then hk is normal if and only if it is selfadjoint; for example, when k ≥ 0,and a normal operator with real spectrum is selfadjoint. In fact,2 ) when h is selfadjoint and k is positive (see [6]; we thank Prof. J. Zemánek for this reference).We note further that, because σ(hk) \ {0} = {w : 0 = w ∈ σ(hk)} = σ((hk) * ) \ {0} = σ(kh) \ {0} = σ(hk) \ {0}, the spectrum σ(hk) is symmetric with respect to the real axis. Conversely, any compact subset K ⊂ C of the complex plane which is symmetric with respect to the real axis can occur as the spectrum σ(hk) for such a pair of operators. To see this, choose a countable sequence (λ n ) n∈N dense in K. The question naturally arises of finding conditions that will guarantee that hk is selfadjoint given that it is normal; we shall answer it in the wider context of Banach algebras.

When Nilpotence Implies the Zeroness of Linear Operators

“…In many results in operator theory, the asymmetric condition σ(A)∩σ(−A) ⊆ {0} yields similar conclusions as when assuming the positivity of A (for instance, it is used in [1] to define the square root of A 2 where A is self-adjoint). It is also known that this asymmetric condition is weaker that positiveness (and negativeness) of A.…”

Unbounded generalizations of the Fuglede-Putnam theorem and applications to the commutativity of self-adjoint operators

“…In [1], [7], [13], [14], [16], [18], [19], [20], [25], and [29], the question of the self-adjointness of the normal product of two self-adjoint operators was tackled in different settings (cf. [2]).…”

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“…In [1], [6], [10], [11], [13], [14], [16], [17], [18], [22], [21], and [25], the question of the self-adjointness of the normal product of two selfadjoint operators was tackled in different settings (cf. [2]).…”