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Register a new userEstimates of Norms on Krein Spaces
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Various norms can be defined on a Krein space by choosing different underlying fundamental decompositions. Some estimates of norms on Krein spaces are discussed and few results in Bognars paper are generalized.
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A definition of frames for Krein spaces is proposed, which extends the notion of $J$-orthonormal basis of Krein spaces. A $J$-frame for a Krein space $(HH, K{,}{,})$ is in particular a frame for $HH$ in the Hilbert space sense. But it is also compatible with the indefinite inner product $K{,}{,}$, meaning that it determines a pair of maximal uniformly $J$-definite subspaces with different positivity, an analogue to the maximal dual pair associated to a $J$-orthonormal basis. Also, each $J$-frame induces an indefinite reconstruction formula for the vectors in $HH$, which resembles the one given by a $J$-orthonormal basis.
The Kahane--Salem--Zygmund inequality is a probabilistic result that guarantees the existence of special matrices with entries $1$ and $-1$ generating unimodular $m$-linear forms $A_{m,n}:ell_{p_{1}}^{n}times cdotstimesell_{p_{m}}^{n}longrightarrowmathbb{R}$ (or $mathbb{C}$) with relatively small norms. The optimal asymptotic estimates for the smallest possible norms of $A_{m,n}$ when $left{ p_{1},...,p_{m}right} subsetlbrack2,infty]$ and when $left{ p_{1},...,p_{m}right} subsetlbrack1,2)$ are well-known and in this paper we obtain the optimal asymptotic estimates for the remaining case: $left{ p_{1},...,p_{m}right} $ intercepts both $[2,infty]$ and $[1,2)$. In particular we prove that a conjecture posed by Albuquerque and Rezende is false and, using a special type of matrices that dates back to the works of Toeplitz, we also answer a problem posed by the same authors.
Let $J$ and $R$ be anti-commuting fundamental symmetries in a Hilbert space $mathfrak{H}$. The operators $J$ and $R$ can be interpreted as basis (generating) elements of the complex Clifford algebra ${mathcal C}l_2(J,R):={span}{I, J, R, iJR}$. An arbitrary non-trivial fundamental symmetry from ${mathcal C}l_2(J,R)$ is determined by the formula $J_{vec{alpha}}=alpha_{1}J+alpha_{2}R+alpha_{3}iJR$, where ${vec{alpha}}inmathbb{S}^2$. Let $S$ be a symmetric operator that commutes with ${mathcal C}l_2(J,R)$. The purpose of this paper is to study the sets $Sigma_{{J_{vec{alpha}}}}$ ($forall{vec{alpha}}inmathbb{S}^2$) of self-adjoint extensions of $S$ in Krein spaces generated by fundamental symmetries ${{J_{vec{alpha}}}}$ (${{J_{vec{alpha}}}}$-self-adjoint extensions). We show that the sets $Sigma_{{J_{vec{alpha}}}}$ and $Sigma_{{J_{vec{beta}}}}$ are unitarily equivalent for different ${vec{alpha}}, {vec{beta}}inmathbb{S}^2$ and describe in detail the structure of operators $AinSigma_{{J_{vec{alpha}}}}$ with empty resolvent set.
We discuss the Krein--von Neumann extensions of three Laplacian-type operators -- on discrete graphs, quantum graphs, and domains. In passing we present a class of one-dimensional elliptic operators such that for any $nin mathbb N$ infinitely many elements of the class have $n$-dimensional null space.
The purpose of this paper is to establish bilinear estimates in Besov spaces generated by the Dirichlet Laplacian on a domain of Euclidian spaces. These estimates are proved by using the gradient estimates for heat semigroup together with the Bony paraproduct formula and the boundedness of spectral multipliers.
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