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It is well known that not all operators satisfy Weyl type theorems simultaneously. In this paper, we denote an operator matrix and consider how eight Weyl type theorems hold for it when certain entry operators satisfy Weyl type theorems. Moreover, we characterize its spectral structure. Finally, the relevant conclusions are promoted to infinite dimensional Hamilton operator.

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-...

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.

In this paper, for given operators A ∈ B ( H ) and B ∈ B ( K ) , we completely describe the set of all C ∈ B ( K , H ) such that M C is bounded below, left (right) semi-Fredholm, respectively.

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 be a separable infinite dimensional Hilbert space, and let and be bounded linear operators on . In this paper, some necessary and sufficient conditions are given for to be invertible for some bounded self-adjoint operator on . Furthermore, as an application of the result above, the perturbation of spectrum for upper triangular Hamiltonian operators is characterized.

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 ) .