By John A De Santo; Conference on Mathematical Methods and Applications of Scattering Theory (1979, Washington, DC)
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Additional info for Mathematical methods and applications of scattering theory : proceedings of a conference held at Cath. Univ. Washington, D.C., May 21-25, 1979
19, etc]. 16]. All other implications required, namely (2) =⇒ (4) =⇒ (3), are trivial. 5. For a Lie subalgebra L of K(X ) the following conditions are equivalent. 1. L is E-solvable. 2. L is triangularizable. 3. [L, L] is Engel. 4. [L, L] is Volterra. 5. The spectral radius is subadditive on ad L. 6. The spectral radius is subadditive on L (that is, ρ (a + b) ≤ ρ (a) + ρ (b) for every a, b ∈ L). An immediate consequence of the above results is that an Engel Lie algebra of compact operators is triangularizable.
6] Let L be a Lie algebra of trace class operators (on a Hilbert space) satisfying tr(ab) = 0 for all a, b ∈ L. Then every finite rank operator in [L, L] is nilpotent. 1) to indicate that tr(ab) = 0 holds for every a ∈ N and b ∈ M , requiring of course that each ab is an operator with trace. 1) is equivalent to the condition tr(c) = 0 for all c ∈ N M . Similarly, we will write N, M = 0 to indicate that a, b = 0 for every a ∈ N and b ∈ M . 2. It should be noted that its proof doesn’t need any modification for operators in a normed operator ideal (see [P]) with spectral trace (recall that the trace is spectral if it coincides with the sum of eigenvalues, taken with multiplicity).
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