Around the Van Daele–Schmüdgen Theorem

Abstract

Abstract. For a bounded non-negative self-adjoint operator acting in a complex, infinitedimensional, separable Hilbert space H and possessing a dense range R we propose a new approach to characterisation of phenomenon concerning the existence of subspaces [36] to refine them. We also develop a new systematic approach, which allows to construct for any unbounded densely defined symmetric/self-adjoint operator T infinitely many pairs T 1 , T 2 of its closed densely defined restrictions

Chernoff-Like Counterexamples Related to Unbounded Operators

“…This result by Schmüdgen (which was generalized later by Brasche and Neidhardt in [2]; see also [1]) is great but remains fairly theoretical. There seems to be no other simple Chernoff-like (whatever simplicity means) example around in the literature except the one by Chernoff.…”

Nonseparability and von Neumann's theorem for domains of unbounded operators

“…Remark 5.7. Several recent results on operator ranges in [AZ15] can immediately be extended to the nonseparable case provided the corresponding operator range satisfies Condition (ii) in Theorem 4.6. In particular, this applies for example to [AZ15, Theorems 3.7, 3.12 and 3.19].…”

On the Kato square root problem

“…Because ran Z 0 = M, ker Z 0 = {0} and ker(I − Z 2 0 ) = {0} we can apply Proposition 4.1 (see statement (5)). So, ran T 2 0 ∩ ran T * 2 0 = {0}.…”