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Register a new userOn the joint spectra of the two dimensional Lie algebra of operators in Hilbert spaces
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Enrico Boasso
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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.
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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.
A bounded linear operator $ A$ on a Hilbert space $ mathcal H $ is said to be an $ EP $ (hypo-$ EP $) operator if ranges of $ A $ and $ A^* $ are equal (range of $ A $ is contained in range of $ A^* $) and $ A $ has a closed range. In this paper, we define $EP$ and hypo-$EP$ operators for densely defined closed linear operators on Hilbert spaces and extend results from bounded operator settings to (possibly unbounded) closed operator settings.
This manuscript introduces a space of functions, termed occupation kernel Hilbert space (OKHS), that operate on collections of signals rather than real or complex functions. To support this new definition, an explicit class of OKHSs is given through the consideration of a reproducing kernel Hilbert space (RKHS). This space enables the definition of nonlocal operators, such as fractional order Liouville operators, as well as spectral decomposition methods for corresponding fractional order dynamical systems. In this manuscript, a fractional order DMD routine is presented, and the details of the finite rank representations are given. Significantly, despite the added theoretical content through the OKHS formulation, the resultant computations only differ slightly from that of occupation kernel DMD methods for integer order systems posed over RKHSs.
A notion of resolvent set for an operator acting in a rigged Hilbert space $D subset Hsubset D^times$ is proposed. This set depends on a family of intermediate locally convex spaces living between $D$ and $D^times$, called interspaces. Some properties of the resolvent set and of the corresponding multivalued resolvent function are derived and some examples are discussed.
Let $G$ be a locally compact abelian group with a Haar measure, and $Y$ be a measure space. Suppose that $H$ is a reproducing kernel Hilbert space of functions on $Gtimes Y$, such that $H$ is naturally embedded into $L^2(Gtimes Y)$ and is invariant under the translations associated with the elements of $G$. Under some additional technical assumptions, we study the W*-algebra $mathcal{V}$ of translation-invariant bounded linear operators acting on $H$. First, we decompose $mathcal{V}$ into the direct integral of the W*-algebras of bounded operators acting on the reproducing kernel Hilbert spaces $widehat{H}_xi$, $xiinwidehat{G}$, generated by the Fourier transform of the reproducing kernel. Second, we give a constructive criterion for the commutativity of $mathcal{V}$. Third, in the commutative case, we construct a unitary operator that simultaneously diagonalizes all operators belonging to $mathcal{V}$, i.e., converts them into some multiplication operators. Our scheme generalizes many examples previously studied by Nikolai Vasilevski and other authors.
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