Weyl ’ S Theorem and Tensor Products : a Counterexample Derek Kitson ,

Approximately fifty percent of Weyl’s theorem fails to transfer from Hilbert space operators to their tensor product. As a biproduct we find that the product of circles in the complex plane is a limaçon. 0. Introduction To say that “Weyl’s theorem holds”, for a bounded operator T ∈ B(X) on a Banach space X , is to assert [2],[4],[5] that 0.1 σ(T ) \ ωess(T ) = π 00 (T ) ≡ iso σ(T )∩π 0 (T ) : the complement in the spectrum of the Weyl spectrum coincides with the isolated eigenvalues of finite multiplicity. One half of this is the assertion that “Browder’s theorem holds” for T ∈ B(X), which is to say [2],[4],[5] that 0.2 ωess(T ) = βess(T ) the Weyl and the Browder spectrum coincide. We would like to apologize for this somewhat anachronistic terminology, which over the years has been allowed to solidify. In this note we offer an example of two operators A,B ∈ B(X), each of which satisfies (0.2), and indeed (0.1), whose tensor product A ⊗ B ∈ B(X) ⊗ B(X) ⊆ B(X ⊗X) does not. We remark that these operators are constructed from rather simple building blocks, as are the resulting spectra. For the record we recall that the Fredholm, Weyl and Browder essential spectrum σess(T ), ωess(T ) and βess(T ) collect complex numbers λ for which T − λI fails to be, respectively, Fredholm, Weyl or Browder; an operator is Fredholm if it has finite dimensional null space and closed range of finite codimension, is Weyl if in addition these two finite dimensions are equal, and is Browder if in addition it has finite ascent and descent [2],[3],[4]. We are writing π(T ) for the eigenvalues of T ∈ B(X), π 0 (T ) for the eigenvalues of finite (geometric) multiplicity and π 00 (T ) for those which are also isolated in the spetrum σ(T ).