example of bounded operator with no eigenvalues

In this entry we show that there are operatorsMathworldPlanetmath with no eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath. Moreover, we exhibit an operator T in a Hilbert spaceMathworldPlanetmath which is bounded, self-adjointPlanetmathPlanetmathPlanetmath, has a non-empty spectrum but no eigenvalues.

Consider the Hilbert space L2([0,1]) (http://planetmath.org/L2SpacesAreHilbertSpaces) and let f:[0,1] be the function f(t)=t.

Let T:L2([0,1])L2([0,1]) be the operator of multiplicationPlanetmathPlanetmath (http://planetmath.org/MultiplicationOperatorOnMathbbL22) by f


Thus, T is a bounded operatorMathworldPlanetmathPlanetmath, since it is a multiplication operator (see this entry (http://planetmath.org/OperatorNormOfMultiplicationOperatorOnL2)). Also, it is easily seen that T is self-adjoint.

We now prove that T has no eigenvalues: suppose λ is an eigenvalue of T and φ is an eigenvectorMathworldPlanetmathPlanetmathPlanetmath. Then,


This means that (f-λ)φ=0, but this is impossible for φ0 since f-λ has at most one zero. Hence, T has no eigenvalues.

Of course, since the Hilbert space is complex, the spectrum of T is non-empty (see this entry (http://planetmath.org/SpectrumIsANonEmptyCompactSet)). Moreover, the spectrum of T can be easily computed and seen to be the whole interval [0,1], as we explain now:

It is known that an operator of multiplication by a continuous functionMathworldPlanetmathPlanetmath g is invertiblePlanetmathPlanetmathPlanetmath if and only if g is invertible. Thus, for every λ, T-λI is easily seen to be the operator of multiplication by (f-λ). Hence, T-λI is not invertible if and only if λ[0,1], i.e. σ(T)=[0,1].

Title example of bounded operator with no eigenvalues
Canonical name ExampleOfBoundedOperatorWithNoEigenvalues
Date of creation 2013-03-22 17:57:53
Last modified on 2013-03-22 17:57:53
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 7
Author asteroid (17536)
Entry type Example
Classification msc 15A18
Classification msc 47A10