A Note on Joint Spectrum in Function Spaces

Given several bounded linear operators $$A_1,..., A_n$$ A 1 , . . . , A n on a Hilbert space, their projective spectrum is the set of complex vectors $$z=(z_1,..., z_n)$$ z = ( z 1 , . . . , z n ) such that the multiparameter pencil $$A(z)=z_1A_1+\cdots +z_nA_n$$ A ( z ) = z 1 A 1 + ⋯ + z n A n is not invertible. This paper studies the projective spectrum of the shift operator T, its adjoint $$T^*$$ T ∗ and a projection operator P. Two spaces of concern are the classical Bergman space $$L_a^2(\mathbb {D})$$ L a 2 ( D ) and the $$L^2$$ L 2 space over the torus $${\mathbb T}^2$$ T 2 . The projective spectra are completely determined in both cases. The results lead to new questions about Toeplitz operators.