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Let H,〈⋅,⋅〉 be a complex Hilbert space and A be a positive bounded linear operator on H. The semi-inner product 〈x,y〉A:=〈Ax,y〉,x,y∈H, induces a semi-norm ∥⋅∥A on H. Let ωA(T) and ∥T∥A denote the A-...

The aim of this article is to study the Fredholm completion problem of two-by-two partial operator matrices in which the lower-left entry is unspecified and others are specified. By using the methods of operator matrix representation and operator equation, we obtain necessity and sufficiency conditions for the partial operator matrices to have a Fredholm completion with the property that the lower-right entry of its Fredholm inverses is specified.

Let H and K be separable infinite dimensional Hilbert spaces, and let A ∈ B ( H ) and B ∈ B ( K ) be given operators. A necessary and sufficient condition is obtained for ( A C 0 B ) to be a CI operator for some (respectively, every) C ∈ B ( K , H ) . Furthermore, some related results are obtained.

Let \({M_{C}}= \left (\begin{array}{ll} A \quad C \\ 0 \quad B \end{array}\right)\) be a 2 × 2 upper triangular operator matrix on Hilbert space \({\mathcal{H} \oplus\mathcal{K}}\) . For given operators \({A\in\mathcal{B}(\mathcal{H})}\) and \({B\in\mathcal{B}(\mathcal{K})}\) , sets \({\bigcap_{C\in\mathcal{B}(\mathcal{K}, \mathcal{H})} \sigma^{g}_{w}(M_{C})}\) and \({\bigcup_{C\in\mathcal{B}(\mathcal{K},\mathcal{H})} \sigma^{g}_{w}(M_{C})}\) are investigated, where \({\sigma^{g}_{w}(\cdot)}\) denotes the generalized Weyl spectrum.

Let \({\mathcal {H}_{1}}\) and \({\mathcal {H}_{2}}\) be separable Hilbert spaces, and let \({A \in \mathcal {B}(\mathcal {H}_{1}),\, B \in \mathcal {B}(\mathcal {H}_{2})}\) and \({C \in \mathcal {B}(\mathcal {H}_{2},\, \mathcal {H}_{1})}\) be given operators. A necessary and sufficient condition is given for \({\left(\begin{smallmatrix}A &\enspace C\\ X &\enspace B \end{smallmatrix}\right)}\) to be a right (left) invertible operator for some \({X \in \mathcal {B}(\mathcal {H}_{1},\, \mathcal {H}_{2})}\). Furthermore, some related results are obtained.

Let H 1 and H 2 be separable infinite-dimensional Hilbert spaces, and let A ∈ B ( H 1 ) , B ∈ B ( H 2 ) and C ∈ B ( H 2 , H 1 ) be given operators. A necessary and sufficient condition is obtained for ( A C X B ) to be a right (left) Fredholm operator for some X ∈ B ( H 1 , H 2 ) .