Browsing by Subject "block operator matrix"

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Access status: Open Access ,
On the eigenvalues of a 2×2 block operator matrix
(2015) Muminov, Muhiddin Èškobilovič; Rasulov, Tulkin Husenovič
A $2×2$ block operator matrix ${\mathbf H}$ acting in the direct sum of one- and two-particle subspaces of a Fock space is considered. The existence of infinitely many negative eigenvalues of $H_{22}$ (the second diagonal entry of ${\mathbf H}$) is proved for the case where both of the associated Friedrichs models have a zero energy resonance. For the number $N(z)$ of eigenvalues of $H_{22}$ lying below z<0, the following asymptotics is found $\lim\limits_{z\to -0} N(z) |\log|z||^{-1}=\,{\mathcal U}_0 \quad (0\lt {\mathcal U}_0\lt \infty).$ Under some natural conditions the infiniteness of the number of eigenvalues located respectively inside, in the gap, and below the bottom of the essential spectrum of ${\mathbf H}$ is proved.
Access status: Open Access ,
Spectrum of J-frame operators
(Wydawnictwa AGH, 2018) Giribet, Juan Ignacio; Langer, Matthias; Leben, Leslie; Maestripieri, Alejandra; Martínez Pería, Francisco; Trunk, Carsten
A $J$-frame is a frame $\mathcal{F}$ for a Krein space $(\mathcal{H},[\cdot,\cdot ])$ which is compatible with the indefinite inner product $[\cdot,\cdot ]$ in the sense that it induces an indefinite reconstruction formula that resembles those produced by orthonormal bases in $\mathcal{H}$. With every $J$-frame the so-called $J$-frame operator is associated, which is a self-adjoint operator in the Krein space $\mathcal{H}$. The $J$-frame operator plays an essential role in the indefinite reconstruction formula. In this paper we characterize the class of $J$-frame operators in a Krein space by a $2\times 2$ block operator representation. The $J$-frame bounds of $\mathcal{F}$ are then recovered as the suprema and infima of the numerical ranges of some uniformly positive operators which are build from the entries of the $2\times 2$ block representation. Moreover, this $2\times 2$ block representation is utilized to obtain enclosures for the spectrum of $J$-frame operators, which finally leads to the construction of a square root. This square root allows a complete description of all $J$-frames associated with a given $J$-frame operator.