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We apply a recent result of Borichev-Golinskii-Kupin on the Blaschke-type conditions for zeros of analytic functions on the complex plane with a cut along the positive semi-axis to the problem of the eigenvalues distribution of the Fredholm-type analytic operator-valued functions.
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P.G. Grinevich L.D. Landau Institute forn Theoretical Physics
, Lomonosov Moscow State University
2015
We construct a Moutard-type transform for the generalized analytic functions. The first theorems and the first explicit examples in this connection are given.
Transposing the Berezin quantization into the setting of analytic microlocal analysis, we construct approximate semiclassical Bergman projections on weighted $L^2$ spaces with analytic weights, and show that their kernel functions admit an asymptotic expansion in the class of analytic symbols. As a corollary, we obtain new estimates for asymptotic expansions of the Bergman kernel on $mathbb{C}^n$ and for high powers of ample holomorphic line bundles over compact complex manifolds.
Let $A$ be a real commutative Banach algebra with unity. Let $a_0in Asetminus{0}$. Let $mathbb Z a_0:={na_0}_{nin mathbb Z}$. Then, $mathbb Z a_0$ is a discrete subgroup of $A$. For any $nin mathbb Z$, the Frechet derivative of the mapping $$x , in , A mapsto x+na_0 , in , A$$ is the identity map on $A$ and, especially, an $A$-linear transformation on $A$. So, the quotient group $A/(mathbb Z a_0)$ is a $1$-dimensional $A$-manifold and the covering projection $$x , in , A mapsto x+mathbb Z a_0 , in , A/(mathbb Z a_0)$$ is an $A$-map. We call $A/(mathbb Z a_0)$ the $1$-dimensional $A$-cylinder by $a_0$. Let $T$ be a compact Hausdorff space. Suppose that there exist $t_1in T$ and $t_2in T$ such that $t_1 ot=t_2$ holds. Then, the set $C(T;mathbb R)$ of all real-valued continuous functions on $T$ is a real commutative Banach algebra with unity and $mathbb R , subsetneq , C(T;mathbb R)$ holds. In this paper, we show that there exists $a_0 , in , C(T;mathbb R)setminus mathbb R$ such that for any $k, in , mathbb N$, the $1$-dimensional $C(T;mathbb R)$-cylinder $(C(T;mathbb R))/(mathbb Z a_0)$ by $a_0$ cannot be embedded in the finite direct product space $(C(T;mathbb R))^k$ as a $C(T;mathbb R)$-submanifold.
In cite{APSIII} Atiyah, Patodi and Singer introduced spectral flow for elliptic operators on odd dimensional compact manifolds. They argued that it could be computed from the Fredholm index of an elliptic operator on a manifold of one higher dimension. A general proof of this fact was produced by Robbin-Salamon cite{RS95}. In cite{GLMST}, a start was made on extending these ideas to operators with some essential spectrum as occurs on non-compact manifolds. The new ingredient introduced there was to exploit scattering theory following the fundamental paper cite{Pu08}. These results do not apply to differential operators directly, only to pseudo-differential operators on manifolds, due to the restrictive assumption that spectral flow is considered between an operator and {its perturbation by a relatively trace-class operator}. In this paper we extend the main results of these earlier papers to spectral flow between an operator and a perturbation satisfying a higher $p^{th}$ Schatten class condition for $0leq p<infty$, thus allowing differential operators on manifolds of any dimension $d<p+1$. In fact our main result does not assume any ellipticity or Fredholm properties at all and proves an operator theoretic trace formula motivated by cite{BCPRSW, CGK16}. This leads us to introduce a notion of `generalised spectral flow for such paths and to investigate its properties.
We prove a sharp Lieb-Thirring type inequality for Jacobi matrices, thereby settling a conjecture of Hundertmark and Simon. An interesting feature of the proof is that it employs a technique originally used by Hundertmark-Laptev-Weidl concerning sums of singular values for compact operators.
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