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namespace OAI
noncomputable section
open scoped InnerProduct
namespace Problem367.OrthogonalBlocks
variable {𝕜 E : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
theorem pow_mem_of_mem
(A : E →L[𝕜] E) (K : Submodule 𝕜 E)
(hAK : ∀ x ∈ K, A x ∈ K) {x : E} (hx : x ∈ K) (n : ℕ) :
(A ^ n) x ∈ K := by
induction n with
| zero => simpa using hx
| succ n ih =>
rw [pow_succ', mul_apply_eq_comp]
exact hAK _ ih
theorem opNorm_le_of_orthogonal_blocks
(A : E →L[𝕜] E) (K : Submodule 𝕜 E) [K.HasOrthogonalProjection]
(hAK : ∀ x ∈ K, A x ∈ K) (hAKo : ∀ x ∈ Kᗮ, A x ∈ Kᗮ)
{c d : ℝ} (hc : 0 ≤ c) (_hd : 0 ≤ d)
(hK : ∀ x ∈ K, ‖A x‖ ≤ c * ‖x‖)
(hKo : ∀ x ∈ Kᗮ, ‖A x‖ ≤ d * ‖x‖) :
‖A‖ ≤ max c d := by
have hM : 0 ≤ max c d := hc.trans (le_max_left _ _)
apply A.opNorm_le_bound hM
intro x
obtain ⟨y, hy, z, hz, rfl⟩ := K.exists_add_mem_mem_orthogonal x
have hyz : inner 𝕜 y z = 0 := Submodule.inner_right_of_mem_orthogonal hy hz
have hAyz : inner 𝕜 (A y) (A z) = 0 :=
Submodule.inner_right_of_mem_orthogonal (hAK y hy) (hAKo z hz)
have hAy : ‖A y‖ ≤ max c d * ‖y‖ :=
(hK y hy).trans (mul_le_mul_of_nonneg_right (le_max_left _ _) (norm_nonneg _))
have hAz : ‖A z‖ ≤ max c d * ‖z‖ :=
(hKo z hz).trans (mul_le_mul_of_nonneg_right (le_max_right _ _) (norm_nonneg _))
apply (sq_le_sq₀ (norm_nonneg _) (mul_nonneg hM (norm_nonneg _))).mp
calc
‖A (y + z)‖ ^ 2 = ‖A y‖ ^ 2 + ‖A z‖ ^ 2 := by
rw [map_add]
simpa only [pow_two] using
norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (A y) (A z) hAyz
_ ≤ (max c d * ‖y‖) ^ 2 + (max c d * ‖z‖) ^ 2 :=
add_le_add
((sq_le_sq₀ (norm_nonneg _) (mul_nonneg hM (norm_nonneg _))).mpr hAy)
((sq_le_sq₀ (norm_nonneg _) (mul_nonneg hM (norm_nonneg _))).mpr hAz)
_ = (max c d * ‖y + z‖) ^ 2 := by
have hnorm : ‖y + z‖ ^ 2 = ‖y‖ ^ 2 + ‖z‖ ^ 2 := by
simpa only [pow_two] using
norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero y z hyz
rw [mul_pow, mul_pow, mul_pow, hnorm]
ring
theorem norm_pow_le_of_orthogonal_blocks
(A : E →L[𝕜] E) (K : Submodule 𝕜 E) [K.HasOrthogonalProjection]
(hAK : ∀ x ∈ K, A x ∈ K) (hAKo : ∀ x ∈ Kᗮ, A x ∈ Kᗮ)
(n : ℕ) {c d : ℝ} (hc : 0 ≤ c) (hd : 0 ≤ d)
(hK : ∀ x ∈ K, ‖(A ^ n) x‖ ≤ c * ‖x‖)
(hKo : ∀ x ∈ Kᗮ, ‖(A ^ n) x‖ ≤ d * ‖x‖) :
‖A ^ n‖ ≤ max c d := by
apply opNorm_le_of_orthogonal_blocks (A ^ n) K
(fun x hx => pow_mem_of_mem A K hAK hx n)
(fun x hx => pow_mem_of_mem A Kᗮ hAKo hx n) hc hd hK hKo
end Problem367.OrthogonalBlocks
end
end OAI
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