On subcorrect and correct pairs of operators in Hilbert spaces (Q1173583)

Let \(\mathcal B(H)\) be the set of all bounded linear operators on the Hilbert space \(\mathcal H\). Let \(A\in{\mathcal B(H)}\) be self-adjoint. Let \[ {\mathcal R}_ +(A)=\{x\in{\mathcal H}\mid\langle Ax,x\rangle\geq 0\}:\quad{\mathcal R}_ - (A)=\{x\in{\mathcal H}\mid\langle Ax,x\rangle\leq 0\}. \] An ordered pair of self-adjoint operators \((A_ 0,A_ 1)\) is called subcorrect if \({\mathcal H}=L_ 0+L_ 1\), where \(L_ 0\), \(L_ 1\) are subspaces of \(\mathcal H\) with \(L_ 0\subseteq{\mathcal R}_ +(A_ 0)\), \(L_ 1\subseteq{\mathcal R}_ -(A_ 1)\). If additionally, \(A_ 0\leq A_ 1\) and if for some \(\alpha,\beta>0\), \(\langle A_ 0 x,x\rangle\geq\alpha\| x\|^ 2\) \((x\in L_ 0)\); \(\langle A_ 1 y,y\rangle\leq-\beta\| y\|^ 2\) \((y\in L_ 1)\), then the pair \((A_ 0,A_ 1)\) is called correct. The authors show that a necessary and sufficient condition for the pair \((A_ 0,A_ 1)\) to be correct is that there exists a continuous map \(A: [0,1]\to{\mathcal B(H)}\) such that for all \(t\) in \([0,1]\), \(A(t)\) is a regular self-adjoint operator with \(A(0)=A_ 0\), \(A(1)=A_ 1\) and \(H_ -(A(t))\subseteq{\mathcal R}_ -(A_ 0)\), where \(H_ -(A(t))\) is the range of \(P_ -(A(t))\) and \[ P_ -(A(t))=\int^ 0_{-\infty} dE_ t(\lambda), \] \(E_ t(\lambda)\) being the spectral-measure function associated with \(A(t)\). An example is given that, even in the case of regular homotopy \(A(t)\), the class of subcorrect pairs is larger than the class of correct ones. The results obtained have applications to problems concerned with periodic Hamiltonian systems.