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Abstract. We consider elementary operators x → n j=1 a j xb j , acting on a unital Banach algebra, where a j and b j are separately commuting families of generalized scalar elements. We give an ascent estimate and a lower bound estimate for such an operator. Additionally, we give a weak variant of the Fuglede-Putnam theorem for an elementary operator with strongly commuting families {a j } and {b j }, i.e. a j = awhere all a 0. Introduction. The theory of generalized scalar operators on a Banach space was developed in [6]. Briefly, a ∈ A is a generalized scalar element of a unital Banach algebra A if it has real spectrum, and if for all real t, e ita ≤ C(1 + |t| s ), for some constant C depending only on a. Also, it is known that these two conditions are equivalent to the existence of a functional calculus for a, based on R. If s = 0, we say that such an element is pre-hermitian. In that case the condition of having real spectrum is not necessary. Also we can define pre-normal elements as elements of the form h+ik with h, k pre-hermitian. Many properties of pre-hermitian, pre-normal, and generalized scalar elements can be found in [6] and [5]. In Section 1 we review results concerning such elements, necessary for reading this note. In [13], a functional calculus for several commuting operators on a Banach space, using Fourier transform, was developed. In Section 2, we prove two results about L 1 behaviour of the Fourier transforms of a family of C ∞ cpt functions. These results have a central role in further applications to the theory of elementary operators on a unital Banach algebra.Section 3 contains applications of the results from Section 2 to elementary operators on a unital Banach algebra A, i.e. to mappings Λ : A → A of the form 2000 Mathematics Subject Classification: 47B48, 47B47, 42B10.

Abstract. We prove the orthogonality of the range and the kernel of an important class of elementary operators with respect to the unitarily invariant norms associated with norm ideals of operators. This class consists of those mappings E : B(H) → B(H), E(X) = AXB + CXD, where B(H) is the algebra of all bounded Hilbert space operators, and A, B, C, D are normal operators, such that AC = CA, BD = DB and ker A∩ker C = ker B ∩ker D = {0}. Also we establish that this class is, in a certain sense, the widest class for which such an orthogonality result is valid. Some other related results are also given.

Abstract. We prove that for Hilbert space operators X and Y , it follows thatRe Y ϕ, Xϕ ,Using the concept of ϕ-Gateaux derivative, we apply this result to characterize orthogonality in the sense of James in B(H), and to give an easy proof of the characterization of smooth points in B(H).