Linear operator on Hilbert space with empty spectrum

[Mathmos6]

Homework Statement

Much as the title says, I need to construct an example of a linear operator on Hilbert space with empty spectrum. I can very easily construct an example with empty point-spectrum (e.g. the right-shift operator on l_2), but this has very far from empty spectrum.

If I recall correctly, any such operator which is also bounded will have non-empty spectrum, so it falls to construct an unbounded example. However, I'm just not really sure how to get going on this. Could anyone help me? Thanks in advance! :)

 

[arkajad]

What is your definition of "an operator on Hilbert space"? Must the domain be dense? What else? What is your definition of the "spectrum"? The devil is in these details.

Reed, "Methods of Modern Mathematical Physics", I. Functional analysis:

 

[Mathmos6]

What is your definition of "an operator on Hilbert space"? Must the domain be dense? What else? What is your definition of the "spectrum"? The devil is in these details.

Reed, "Methods of Modern Mathematical Physics", I. Functional analysis:

My definition of a Hilbert space is standard, i.e. a real or complex inner product space which is complete under the norm defined by the inner product. The spectrum of an operator T is, for me, the set of points 'p' in the complex plane for which T-pI is not invertible (I the identity map). I have never really looked at operators which are not on the whole space before - so though it was never explicitly stated, I believe the operator should have domain the entire Hilbert space (whichever Hilbert space it may be) - is such a thing possible? The wording of the question makes it seem to me like we aren't expected to state such an operator but merely to provide a means to construct one - though of course, if you can state one explicitly that's definitely better!

I am nowhere near a library which has any decent math texts at the moment unfortunately, but as soon as I get near one again I will be sure to track it down, thankyou!

 

[arkajad]

Alright I am attaching the relevant page. Part of the example (that T2 is closed) is left as an exercise. But this example may help you to construct your own example.