Let G be a locally compact
group with continuous unitary representations R_{i} acting on the Hilbert spaces
H(R_{t}),i = 1,2. Suppose that R_{2} is irreducible. A closed subspace of H(R_{1}), called
the null space coming from R_{1} at R_{2} and denoted F(R_{1};R_{2}), is defined. Write ℋ^{c}
for the conjugate space of the Hilbert space ℋ. The following theorem is
proved.
Theorem 1. Let G be a compact group with closed subgroup H. Let M,L be
irreducible unitary representations of G,H, respectively. Let U^{L} be the induced
representation of L, and let M_{E} be the restriction of M to H. Then the following are
equivalent:
(a) The classical Frobenius Reciprocity Theorem.
(b) H(M) ⊗ F(M_{II};L)^{c}≅F(U^{L};M) ⊗ H(L)^{c}.
When G is not compact, both (a) and (b) may fail. A nonHilbert Banach space
induced representation (W^{L}) is defined. Let G be a locally compact group with
closed subgroup H. Let M,L be irreducible unitary representations of G,H,
respectively, where H(L) is separable. Spaces F_{0}(W^{L};M) ⊗^{∗}H(L)^{c} (shown to equal
F(U^{L};M) ⊗ H(L)^{c} when G is compact) and QF(M_{E};L) shown to equal
F(M_{H};L) when L is compact) are defined. The following generalization of (b) is
shown.
Theorem 2. Let G,H,L be as above. Then
