Spectral Shorted Operators |
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Authors: | Jorge Antezana Gustavo Corach Demetrio Stojanoff |
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Institution: | (1) Departamento de Matemática Facultad de Ciencias Exactas, Universidad Nacional de La Plata, IAM-CONICET, 50 y 115, 1900 La Plata, Argentina;(2) Departamento de Matemática, Facultad de Ingeniería-UBA, IAM-CONICET, Av. Paseo Colón 850 (1063), Buenos Aires, Argentina;(3) Departamento de Matemática Facultad de Ciencias Exactas, Universidad Nacional de La Plata, IAM-CONICET, 50 y 115, 1900 La Plata, Argentina |
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Abstract: | If
$$\mathcal{H}$$ is a Hilbert space,
$$\mathcal{S}$$ is a closed subspace of
$$\mathcal{H},$$ and A is a positive bounded linear operator on
$$\mathcal{H},$$ the spectral shorted operator
$$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ is defined as the infimum of the sequence
$$\sum (\mathcal{S},A^n )^{1/n} ,$$ where denotes
$$\sum \left( {\mathcal{S},B} \right)$$ the shorted operator of B to
$$\mathcal{S}.$$ We characterize the left spectral resolution of
$$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ and show several properties of this operator, particularly in the case that
dim
$${\mathcal{S} = 1.}$$ We use these results to generalize the concept of Kolmogorov complexity for the infinite dimensional
case and for non invertible operators. |
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Keywords: | 47A30 47B15 |
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