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Register a new userOn selfadjoint extensions of symmetric operator with exit from space
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We investigate minimal operator corresponding to operator differential expression with exit from space, study its selfadjoint extensions, also for one particular selfadjoint extension corresponding to boundary value problem with some rational function of eigenparameter in boundary condition establish asymptotics of spectrum and derive trace formula
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We study the problem of finding the instability index of certain non-selfadjoint fourth order differential operators that appear as linearizations of coating and rimming flows, where a thin layer of fluid coats a horizontal rotating cylinder. The main result reduces the computation of the instability index to a finite-dimensional space of trigonometric polynomials. The proof uses Lyapunovs method to associate the differential operator with a quadratic form, whose maximal positive subspace has dimension equal to the instability index. The quadratic form is given by a solution of Lyapunovs equation, which here takes the form of a fourth order linear PDE in two variables. Elliptic estimates for the solution of this PDE play a key role. We include some numerical examples.
In this note we define and study a Hilbert space-valued stochastic integral of operator-valued functions with respect to Hilbert space-valued measures. We show that this integral generalizes the classical Ito stochastic integral of adapted processes with respect to normal martingales and the Ito integral in a Fock space
Given a densely defined and closed operator $A$ acting on a complex Hilbert space $mathcal{H}$, we establish a one-to-one correspondence between its closed extensions and subspaces $mathfrak{M}subsetmathcal{D}(A^*)$, that are closed with respect to the graph norm of $A^*$ and satisfy certain conditions. In particular, this will allow us to characterize all densely defined and closed restrictions of $A^*$. After this, we will express our results using the language of Gelfand triples generalizing the well-known results for the selfadjoint case. As applications we construct: (i) a sequence of densely defined operators that converge in the generalized sense to a non-densely defined operator, (ii) a non-closable extension of a symmetric operator and (iii) selfadjoint extensions of Laplacians with a generalized boundary condition.
In this note, we consider the Dirac operator $D$ on a Riemannian symmetric space $M$ of noncompact type. Using representation theory we show that $D$ has point spectrum iff the $hat A$-genus of its compact dual does not vanish. In this case, if $M$ is irreducible then $M = U(p,q)/U(p) times U(q)$ with $p+q$ odd, and $Spec_p(D) = {0}$.
The main object of the paper is a symmetric system $J y-B(t)y=lD(t) y$ defined on an interval $cI=[a,b) $ with the regular endpoint $a$. Let $f(cd,l)$ be a matrix solution of this system of an arbitrary dimension and let $(Vf)(s)=intlimits_cI f^*(t,s)D(t)f(t),dt$ be the Fourier transform of the function $f(cd)in L_D^2(cI)$. We define a pseudospectral function of the system as a matrix-valued distribution function $s(cd)$ of the dimension $n_s$ such that $V$ is a partial isometry from $L_D^2(cI)$ to $L^2(s;bC^{n_s})$ with the minimally possible kernel. Moreover, we find the minimally possible value of $n_s$ and parameterize all spectral and pseudospectral functions of every possible dimensions $n_s$ by means of a Nevanlinna boundary parameter. The obtained results develop the results by Arov and Dym; A.~Sakhnovich, L.~Sakhnovich and Roitberg; Langer and Textorius.
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