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We mainly deal with the relationship between the local spectrum as well as propery ( $$\beta $$ ) and decomposability of certain $$2\times 2$$ operator matrices with that of its individual entries. As application of these results, we obtain several relevant conclusions of Hamilton type operator.

In this paper, for given operators $A\in\B(\H)$ and $B\in\B(\K)$, the sets of all $C\in \B(\K,\H)$ such that $M_C=\bmatrix{cc} A&C\\0&B\endbmatrix$ is generalized Weyl and generalized left (right) Weyl, are completely described. Furthermore, the following intersections and unions of the generalized left Weyl spectra $$ \bigcup_{C\in\B(\K,\H)}\sigma^g_{lw}(M_C) \;\;\; \mbox{and} \;\;\; \bigcap_{C\in\B(\K,\H)}\sigma^g_{lw}(M_C) $$ are also described, and necessary and sufficient conditions which two operators $A\in\B(\H)$ and $B\in\B(\K)$ have to satisfy in order for $M_C$ to be a generalized left Weyl operator for each $C\in\B(\K,\H)$, are presented.

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 \(\mathcal X \) and \(\mathcal Y \) be Banach spaces, and let \(A\in \mathcal B (\mathcal X )\) and \(C\in \mathcal B (\mathcal Y , \mathcal X )\) be given operators. A necessary and sufficient condition is given for \(\left[ \begin{array}{cc} A&C \\ X&Y \\ \end{array} \right]\) to be invertible (respectively, left invertible) for some \(X\in \mathcal B (\mathcal X , \mathcal Y )\) and \(Y\in \mathcal B (\mathcal Y )\). Furthermore, some related results are obtained.

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 ) .