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- W4246154932 abstract "We establish several inequalities for eigenvalues of selfadjoint operators in Hilbert space. The results are quite general. In particular, let $Omega$ be a region in ${{mathbf {R}}^n},partial Omega$ its boundary and $Delta$ the Laplace operator in ${{mathbf {R}}^n}$. Let $p(x)$ be a polynomial of degree $m$ having nonnegative real coefficients. We show that if the problems (1) $- Delta u = lambda u$ in $Omega ;u = 0$ on $partial Omega$; (2) $p( - Delta )upsilon = mu upsilon$ in $Omega ;upsilon$ and its first $m - 1 text {derivatives}=0 text {on} partial Omega$; and (3) ${( - Delta )^m}w = vw$ in $Omega ;w$ and its first $m - 1 text {derivatives}=0 text {on} partial Omega$ are selfadjoint with discrete spectra of finite multiplicity ${lambda _1} leq {lambda _2} leq cdots$ etc. then (4) $p(Gamma _i^{1/m}) geq {mu _i} geq p({lambda _i})$ for each index $i$. The set of problems (1), (2), (3) and the result (4) is only one example of our more general result. The above problems (1), (2), and (3) can be thought of as related through the single operator given by the Laplacian. We also establish results for eigenvalues for unrelated operators. Let $A$, $B$ and $A + B$ be selfadjoint on domains ${D_A},{D_B}$, and ${D_{A + B}}$ with ${D_{A + B}} subseteq {D_A} cap {D_B}$. If $A$, $B$, and $A + B$ have discrete spectra ${ {lambda _i}} _{i = 1}^infty ,{ {mu _i}} _{i = 1}^infty$ and ${ {Gamma _i}} _{i = 1}^infty$ arranged in ascending order, as above, then inequality (5) $sum nolimits _{i = 1}^n {{Gamma _i}} geq sum nolimits _{i = 1}^n {({lambda _i} + {v_i})}$ is established for each positive integer $n$." @default.
- W4246154932 created "2022-05-12" @default.
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- W4246154932 date "1990-03-01" @default.
- W4246154932 modified "2023-10-17" @default.
- W4246154932 title "Inequalities for Eigenvalues of Selfadjoint Operators" @default.
- W4246154932 doi "https://doi.org/10.2307/2001237" @default.
- W4246154932 hasPublicationYear "1990" @default.
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