Index: lldb/source/Plugins/Language/CPlusPlus/CPlusPlusNameParser.cpp
===================================================================
--- lldb/source/Plugins/Language/CPlusPlus/CPlusPlusNameParser.cpp
+++ lldb/source/Plugins/Language/CPlusPlus/CPlusPlusNameParser.cpp
@@ -329,6 +329,24 @@
   }
 
   const auto &token = Peek();
+  if (token.getKind() == tok::lessless) {
+    if (m_next_token_index + 1 < m_tokens.size()) {
+      clang::Token n_token = m_tokens[m_next_token_index + 1];
+      if (n_token.getKind() != tok::l_paren && n_token.getKind() != tok::less) {
+        clang::Token tmp_tok;
+
+        tmp_tok.setLength(1);
+        tmp_tok.setLocation(token.getLocation().getLocWithOffset(1));
+        tmp_tok.setKind(tok::less);
+
+        m_tokens[m_next_token_index] = tmp_tok;
+
+        start_position.Remove();
+        return true;
+      }
+    }
+  }
+
   switch (token.getKind()) {
   case tok::kw_new:
   case tok::kw_delete:
Index: lldb/unittests/Language/CPlusPlus/CPlusPlusLanguageTest.cpp
===================================================================
--- lldb/unittests/Language/CPlusPlus/CPlusPlusLanguageTest.cpp
+++ lldb/unittests/Language/CPlusPlus/CPlusPlusLanguageTest.cpp
@@ -140,12 +140,20 @@
        "std::vector<Class, std::allocator<Class>>",
        "_M_emplace_back_aux<Class const&>"},
       {"`anonymous namespace'::foo", "`anonymous namespace'", "foo"},
-      {"`operator<<A>'::`2'::B<0>::operator>",
-       "`operator<<A>'::`2'::B<0>",
+      {"`operator<<A>'::`2'::B<0>::operator>", "`operator<<A>'::`2'::B<0>",
        "operator>"},
       {"`anonymous namespace'::S::<<::__l2::Foo",
-       "`anonymous namespace'::S::<<::__l2",
-       "Foo"}};
+       "`anonymous namespace'::S::<<::__l2", "Foo"},
+      {"A::operator><A::B>", "A", "operator><A::B>"},
+      {"operator><A::B>", "", "operator><A::B>"},
+      {"A::operator<<A::B>", "A", "operator<<A::B>"},
+      {"operator<<A::B>", "", "operator<<A::B>"},
+      {"A::operator<<<A::B>", "A", "operator<<<A::B>"},
+      {"operator<<<A::B>", "", "operator<<<A::B>"},
+      // We expect these cases to fail until we turn on C++2a
+      //      {"A::operator<=><A::B>", "A", "operator<=><A::B>"},
+      //      {"operator<=><A::B>", "", "operator<=><A::B>" },
+  };
 
   llvm::StringRef context, basename;
   for (const auto &test : test_cases) {