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Register a new userJoint spectrum for quasi-solvable Lie algebras of operators
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Authors
Enrico Boasso
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Given a complex Banach space $X$ and a joint spectrum for complex solvable finite dimensional Lie algebras of operators defined on $X$, we extend this joint spectrum to quasi-solvable Lie algebras of operators, and we prove the main spectral properties of the extended joint spectrum. We also show that this construction is uniquely determined by the original joint spectrum.
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Given two complex Banach spaces $X_1$ and $X_2$, a tensor product $X_1tilde{otimes} X_2$ of $X_1$ and $X_2$ in the sense of [14], two complex solvable finite dimensional Lie algebras $L_1$ and $L_2$, and two representations $rho_icolon L_ito {rm L}(X_i)$ of the algebras, $i=1$, $2$, we consider the Lie algebra $L=L_1times L_2$, and the tensor product representation of $L$, $rhocolon Lto {rm L}(X_1tilde{otimes}X_2)$, $rho=rho_1otimes I +Iotimes rho_2$. In this work we study the S{l}odkowski and the split joint spectra of the representation $rho$, and we describe them in terms of the corresponding joint spectra of $rho_1$ and $rho_2$. Moreover, we study the essential S{l}odkowski and the essential split joint spectra of the representation $rho$, and we describe them by means of the corresponding joint spectra and the corresponding essential joint spectra of $rho_1$ and $rho_2$. In addition, with similar arguments we describe all the above-mentioned joint spectra for the multiplication representation in an operator ideal between Banach spaces in the sense of [14]. Finally, we consider nilpotent systems of operators, in particular commutative, and we apply our descriptions to them.
In this paper solvable Leibniz algebras whose nilradical is quasi-filiform Lie algebra of maximum length, are classified. The rigidity of such Leibniz algebras with two-dimensional complemented space to nilradical is proved.
We show that in the class of solvable Lie algebras there exist algebras which admit local derivations which are not ordinary derivation and also algebras for which every local derivation is a derivation. We found necessary and sufficient conditions under which any local derivation of solvable Lie algebras with abelian nilradical and one-dimensional complementary space is a derivation. Moreover, we prove that every local derivation on a finite-dimensional solvable Lie algebra with model nilradical and maximal dimension of complementary space is a derivation.
We consider the complex solvable non-commutative two dimensional Lie algebra $L$, $L=<y>oplus <x>$, with Lie bracket $[x,y]=y$, as linear bounded operators acting on a complex Hilbert space $H$. Under the assumption $R(y)$ closed, we reduce the computation of the joint spectra $Sp(L,E)$, $sigma_{delta ,k}(L,E)$ and $sigma_{pi ,k}(L,E)$, $k= 0,1,2$, to the computation of the spectrum, the approximate point spectrum, and the approximate compression spectrum of a single operator. Besides, we also study the case $y^2=0$, and we apply our results to the case $H$ finite dimensional.
We give a necessary and sufficient condition for amenability of the Banach algebra of approximable operators on a Banach space. We further investigate the relationship between amenability of this algebra and factorization of operators, strengthening known results and developing new techniques to determine whether or not a given Banach space carries an amenable algebra of approximable operators. Using these techniques, we are able to show, among other things, the non-amenability of the algebra of approximable operators on Tsirelsons space.
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