Unital Compact Homomorphisms between Extended Analytic Lipschitz Algebras

Let ð ‘‹ and ð ¾ be compact plane sets with ð ¾ âŠ† ð ‘‹ . We define ð ´ ( ð ‘‹ , ð ¾ ) = { ð ‘“ ∈ ð ¶ ( ð ‘‹ ) ∶ ð ‘“ | ð ¾ âˆˆ ð ´ ( ð ¾ ) } , where ð ´ ( ð ¾ ) = { ð ‘” ∈ ð ¶ ( ð ‘‹ ) ∶ ð ‘” is analytic on i n t ( ð ¾ ) } . For ð ›¼ ∈ ( 0 , 1 ] , we define L i p ( ð ‘‹ , ð ¾ , ð ›¼ ) = { ð ‘“ ∈ ð ¶ ( ð ‘‹ ) ∶ ð ‘ ð ›¼ , ð ¾ ( ð ‘“ ) = s u p { | ð ‘“ ( 𠑧 ) − ð ‘“ ( 𠑤 ) | / | 𠑧 − 𠑤 | ð ›¼ ∶ 𠑧 , 𠑤 ∈ ð ¾ , 𠑧 ≠𠑤 } < ∞ } and L i p ð ´ ( ð ‘‹ , ð ¾ , ð ›¼ ) = ð ´ ( ð ‘‹ , ð ¾ ) ∩ L i p ( ð ‘‹ , ð ¾ , ð ›¼ ) . It is known that L i p ð ´ ( ð ‘‹ , ð ¾ , ð ›¼ ) is a natural Banach function algebra on ð ‘‹ under the norm | | ð ‘“ | | L i p ( ð ‘‹ , ð ¾ , ð ›¼ ) = | | ð ‘“ | | ð ‘‹ + ð ‘ ð ›¼ , ð ¾ ( ð ‘“ ) , where | | ð ‘“ | | ð ‘‹ = s u p { | ð ‘“ ( ð ‘¥ ) | ∶ ð ‘¥ ∈ ð ‘‹ } . These algebras are called extended analytic Lipschitz algebras. In this paper we study unital homomorphisms from natural Banach function subalgebras of L i p ð ´ ( ð ‘‹ 1 , ð ¾ 1 , ð ›¼ 1 ) to natural Banach function subalgebras of L i p ð ´ ( ð ‘‹ 2 , ð ¾ 2 , ð ›¼ 2 ) and investigate necessary and sufficient conditions for which these homomorphisms are compact. We also determine the spectrum of unital compact endomorphisms of L i p ð ´ ( ð ‘‹ , ð ¾ , ð ›¼ ) .