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تسجيل مستخدم جديدSpectral averaging for trace compatible operators
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In this note the notions of trace compatible operators and infinitesimal spectral flow are introduced. We define the spectral shift function as the integral of infinitesimal spectral flow. It is proved that the spectral shift function thus defined is absolutely continuous and Kreins formula is established. Some examples of trace compatible affine spaces of operators are given.
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For a commuting $d$- tuple of operators $boldsymbol T$ defined on a complex separable Hilbert space $mathcal H$, let $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ be the $dtimes d$ block operator $big (!!big (big [ T_j^* , T_ibig ]big )!!
big )$ of the commutators $[T^*_j , T_i] := T^*_j T_i - T_iT_j^*$. We define the determinant of $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ by symmetrizing the products in the Laplace formula for the determinant of a scalar matrix. We prove that the determinant of $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ equals the generalized commutator of the $2d$ - tuple of operators, $(T_1,T_1^*, ldots, T_d,T_d^*)$ introduced earlier by Helton and Howe. We then apply the Amitsur-Levitzki theorem to conclude that for any commuting $d$ - tuple of $d$ - normal operators, the determinant of $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ must be $0$. We show that if the $d$- tuple $boldsymbol T$ is cyclic, the determinant of $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ is non-negative and the compression of a fixed set of words in $T_j^* $ and $T_i$ -- to a nested sequence of finite dimensional subspaces increasing to $mathcal H$ -- does not grow very rapidly, then the trace of the determinant of the operator $big [!! big [ boldsymbol T^* , boldsymbol Tbig ] !!big ]$ is finite. Moreover, an upper bound for this trace is given. This upper bound is shown to be sharp for a class of commuting $d$ - tuples. We make a conjecture of what might be a sharp bound in much greater generality and verify it in many examples.
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