Orthogonality relations for extremal vectors
by
Alexandru Terescenco
West University Timisoara

The extremal vectors were introduced by Ansari and Enflo. Let H be a Hilbert space, T a linear bounded operator on H with dense range, x₀ ∈ H and e ∈ (0, ∥x₀∥) . The backward extremal vector y_e is the unique vector of minimal norm in {y:∥Ty-x₀∥ ≤ e} . With H, x₀ and e as above, but with T one-to-one instead of having dense range, the forward extremal vector v_e is the unique vector in {v:∥v-x₀∥ ≤ e} such that ∥Tv∥ is minimal. The sequence y_eⁿ (or v_eⁿ) of extremal vectors associated to Tⁿ, n=1, 2, ... proved to be useful for finding nontrivial hyperinvariant subspaces for T (assuming that T has some additional properties). A key feature of the extremal vectors is the orthogonality relation:

r⊥ye⇔ x0-Tye⊥Tr or x0-ve⊥r⇔Tve⊥Tr

We propose a general framework for extremal vectors suggested by interpolation theory. Suppose H=Y+Z, with Y, Z operator ranges (with range norm). The Gagliardo diagram of x₀ ∈ H is the convex set

G(x0)={(t, s) ∈ R2+:(∃)y ∈ Y, (∃) z ∈ Z   such that  x0=y+z, ∥y∥Y ≤ t, ∥z∥Z ≤ s}

For (t, s) ∈ ∂G(x₀) there is a unique minimal decomposition x₀=y_t+z_s,   ∥y_t∥_Y=t,   ∥z_s∥_Z=s and the orthogonality relation

r⊥yt in Y⇔ r⊥zs in Z

holds for r ∈ Y∩Z. The orthogonality relations for backward and forward extremal vectors are obtained considering G(x₀) and G(Tx₀) respectively, in H= range (T)+H.

Date received: June 16, 2008