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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).

Morlet and T. Paul, Transforms associated to square integrable group representations II: Examples, Ann. Inst. Henri Poincar´e 45(1986), 293–309. [9] S. Grudsky, A. Karapetyants and N. Vasilevski, Dynamics of properties of Toeplitz operators on the upper half-plane: Parabolic case, J. Operator Theory 52(2004), 185– 204. 46 Ondrej Hutn´ık IEOT [10] O. Hutn´ık, A note on wavelet subspaces (submitted). [11] Q. Jiang and L. Z. Peng, Toeplitz and Hankel type operators on the upper half-plane, Integral Equations Operator Theory 15(1992), 744–767.